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Search: a193649 -id:a193649
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A193722
Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.
+10
93
1, 1, 2, 1, 5, 6, 1, 8, 21, 18, 1, 11, 45, 81, 54, 1, 14, 78, 216, 297, 162, 1, 17, 120, 450, 945, 1053, 486, 1, 20, 171, 810, 2295, 3888, 3645, 1458, 1, 23, 231, 1323, 4725, 10773, 15309, 12393, 4374, 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553, 13122
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials
...
q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k),
...
for k=0,1,2,... The Q-upstep of p is the polynomial given by
...
U(p) = p(n)*q(n+1,x) + p(n-1)*q(n,x) + ... + p(0)*q(1,x); note that q(0,x) does not appear.
...
Now suppose that P=(p(n,x)) and Q=(q(n,x)) are sequences of polynomials, where n indicates degree. The fusion of P by Q, denoted by P**Q, is introduced here as the sequence W=(w(n,x)) of polynomials defined by w(0,x)=1 and w(n+1,x)=U(p(n,x)).
...
Strictly speaking, ** is an operation on sequences of polynomials. However, if P and Q are regarded as numerical triangles (e.g., coefficients of polynomials), then ** can be regarded as an operation on numerical triangles. In this case, row (n+1) of P**Q, for n >= 0, is given by the matrix product P(n)*QQ(n), where P(n)=(p(n,n)...p(n,n-1)......p(n,1), p(n,0)) and QQ(n) is the (n+1)-by-(n+2) matrix given by
...
q(n+1,0) .. q(n+1,1)........... q(n+1,n) .... q(n+1,n+1)
0 ......... q(n,0)............. q(n,n-1) .... q(n,n)
0 ......... 0.................. q(n-1,n-2) .. q(n-1,n-1)
...
0 ......... 0.................. q(2,1) ...... q(2,2)
0 ......... 0 ................. q(1,0) ...... q(1,1);
here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x+q(k,k-1); i.e., "q" is used instead of "t".
...
If s=(s(1),s(2),s(3),...) is a sequence, then the infinite square matrix indicated by
s(1)...s(2)...s(3)...s(4)...s(5)...
..0....s(1)...s(2)...s(3)...s(4)...
..0......0....s(1)...s(2)...s(3)...
..0......0.......0...s(1)...s(2)...
is the self-fusion matrix of s; e.g., A202453, A202670.
...
Example: let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n. Then
...
w(0,x) = 1 by definition of W
w(1,x) = U(p(0,x)) = U(1) = p(0,0)*q(1,x) = 1*(x+2) = x+2;
w(2,x) = U(p(1,x)) = U(x+1) = q(2,x) + q(1,x) = x^2+5x+6;
w(3,x) = U(p(2,x)) = U(x^2+2x+1) = q(3,x) + 2q(2,x) + q(1,x) = x^3+8x^2+21x+18;
...
From these first 4 polynomials in the sequence P**Q, we can write the first 4 rows of P**Q when P, Q, and P**Q are regarded as triangles:
1;
1, 2;
1, 5, 6;
1, 8, 21, 18;
...
Generally, if P and Q are the sequences given by p(n,x)=(ax+b)^n and q(n,x)=(cx+d)^n, then P**Q is given by (cx+d)(bcx+a+bd)^n.
...
In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.
...
..P...........Q.........P**Q.......r(P**Q)
(x+1)^n.....(x+1)^n.....A081277....A118800 (unsigned)
(x+1)^n.....(x+2)^n.....A193722....A193723
(x+2)^n.....(x+1)^n.....A193724....A193725
(x+2)^n.....(x+2)^n.....A193726....A193727
(x+2)^n.....(2x+1)^n....A193728....A193729
(2x+1)^n....(x+1)^n.....A038763....A136158
(2x+1)^n....(2x+1)^n....A193730....A193731
(2x+1)^n,...(x+1)^n.....A193734....A193735
...
Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.
...
P.............Q.........P**Q.......r(P**Q)
Fib[n+1,x]...(x+1)^n....A193736....A193737
u.............u.........A193738....A193739
u**u..........u**u......A193740....A193741
...
Regarding A193722:
col 1 ..... A000012
col 2 ..... A016789
col 3 ..... A081266
w(n,n) .... A025192
w(n,n-1) .. A081038
...
Associated with "upstep" as defined above is "downstep" defined at A193842 in connection with fission.
LINKS
Jinyuan Wang, Rows n=0..101 of triangle, flattened
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
FORMULA
Triangle T(n,k), read by rows, given by [1,0,0,0,0,0,0,0,...] DELTA [2,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
T(n,k) = 3*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
T(n, k) = 3^(k-1)*( binomial(n-1,k) + 2*binomial(n,k) ). - G. C. Greubel, Feb 18 2020
EXAMPLE
First six rows:
1;
1, 2;
1, 5, 6;
1, 8, 21, 18;
1, 11, 45, 81, 54;
1, 14, 78, 216, 297, 162;
MAPLE
fusion := proc(p, q, n) local d, k;
p(n-1, 0)*q(n, x)+add(coeff(p(n-1, x), x^k)*q(n-k, x), k=1..n-1);
[1, seq(coeff(%, x, n-1-k), k=0..n-1)] end:
p := (n, x) -> (x + 1)^n; q := (n, x) -> (x + 2)^n;
A193722_row := n -> fusion(p, q, n);
for n from 0 to 5 do A193722_row(n) od; # Peter Luschny, Jul 24 2014
MATHEMATICA
(* First program *)
z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)
(* Second program *)
Table[3^(k-1)*(Binomial[n-1, k] +2*Binomial[n, k]), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)
PROG
(Sage)
def fusion(p, q, n):
F = p(n-1, 0)*q(n, x)+add(expand(p(n-1, x)).coefficient(x, k)*q(n-k, x) for k in (1..n-1))
return [1]+[expand(F).coefficient(x, n-1-k) for k in (0..n-1)]
A193842_row = lambda k: fusion(lambda n, x: (x+1)^n, lambda n, x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 24 2014
(PARI) T(n, k) = 3^(k-1)*(binomial(n-1, k) +2*binomial(n, k)); \\ G. C. Greubel, Feb 18 2020
(Magma) [3^(k-1)*( Binomial(n-1, k) + 2*Binomial(n, k) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020
(GAP) Flat(List([0..10], n-> List([0..n], k-> 3^(k-1)*( Binomial(n-1, k) + 2*Binomial(n, k) ) ))); # G. C. Greubel, Feb 18 2020
CROSSREFS
Cf. A081277, A084938, A118800, A193649, A193723-A193741, A202453.
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 04 2011
STATUS
approved
A193842
Triangular array: the fission of the polynomial sequence ((x+1)^n: n >= 0) by the polynomial sequence ((x+2)^n: n >= 0). (Fission is defined at Comments.)
+10
27
1, 1, 4, 1, 7, 13, 1, 10, 34, 40, 1, 13, 64, 142, 121, 1, 16, 103, 334, 547, 364, 1, 19, 151, 643, 1549, 2005, 1093, 1, 22, 208, 1096, 3478, 6652, 7108, 3280, 1, 25, 274, 1720, 6766, 17086, 27064, 24604, 9841, 1, 28, 349, 2542, 11926, 37384, 78322, 105796
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials:
...
q(k,x) = t(k,0)*x^k + t(k,1)*x^(k-1) + ... + t(k,k-1)*x + t(k,k),
...
for k = 0, 1, 2, ... The Q-downstep of p is the polynomial given by
...
D(p) = p(n)*q(n-1,x) + p(n-1)*q(n-2,x) + ... + p(1)*q(0,x). (Note that p(0) does not appear. "Q-downstep" as just defined differs slightly from "Q-downstep" as defined for a different purpose at A193649.)
...
Now suppose that P = (p(n,x): n >= 0) and Q = (q(n,x): n >= 0) are sequences of polynomials, where n indicates degree. The fission of P by Q, denoted by P^^Q, is introduced here as the sequence W = (w(n,x): n >= 0) of polynomials defined by w(0,x) = 1 and w(n,x) = D(p(n+1,x)).
...
Strictly speaking, ^^ is an operation on sequences of polynomials. However, if P and Q are regarded as numerical triangles (of coefficients of polynomials), then ^^ can be regarded as an operation on numerical triangles. In this case, row n of P^^Q, for n > 0, is given by the matrix product P(n+1)*QQ(n), where P(n+1) =(p(n+1,n+1), p(n+1,n), ..., p(n+1,2), p(n+1,1)) and QQ(n) is the (n+1)-by-(n+1) matrix given by
...
q(n,0) .. q(n,1)............. q(n,n-1) .... q(n,n)
0 ....... q(n-1,0)........... q(n-1,n-2)... q(n-1,n-1)
0 ....... 0.................. q(n-2,n-3) .. q(n-2,n-2)
...
0 ....... 0.................. q(1,0) ...... q(1,1)
0 ....... 0 ................. 0 ........... q(0,0).
Here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x + q(k,k);
i.e., "q" is used instead of "t".
...
Example: Let p(n,x) = (x+1)^n and q(n,x) = (x+2)^n. Then
...
w(0,x) = 1 by the definition of W,
w(1,x) = D(p(2,x)) = 1*(x+2) + 2*1 = x + 4,
w(2,x) = D(p(3,x)) = 1*(x^2+4*x+4) + 3*(x+2) + 3*1 = x^2 + 7*x + 13,
w(3,x) = D(p(4,x)) = 1*(x^3+6*x^2+12*x+8) + 4*(x^2+4x+4) + 6*(x+2) + 4*1 = x^3 + 10*x^2 + 34*x + 40.
...
From these first 4 polynomials in the sequence P^^Q, we can write the first 4 rows of P^^Q when P, Q, and P^^Q are regarded as triangles:
1
1...4
1...7....13
1...10...34...40
...
In the following examples, r(P^^Q) is the mirror of P^^Q, obtained by reversing the rows of P^^Q. Let u denote the polynomial x^n + x^(n-1) + ... + x + 1.
...
..P........Q...........P^^Q........r(P^^Q)
(x+1)^n....(x+2)^n.....A193842.....A193843
(x+1)^n....(x+1)^n.....A193844.....A193845
(x+2)^n....(x+1)^n.....A193846.....A193847
(2x+1)^n...(x+1)^n.....A193856.....A193857
(x+1)^n....(2x+1)^n....A193858.....A193859
(x+1)^n.......u........A054143.....A104709
..u........(x+1)^n.....A074909.....A074909
..u...........u........A002260.....A004736
(x+2)^n.......u........A193850.....A193851
..u.........(x+2)^n....A193844.....A193845
(2x+1)^n......u........A193860.....A193861
..u.........(2x+1)^n...A115068.....A193862
...
Regarding A193842,
col 1 ...... A000012
col 2 ...... A016777
col 3 ...... A081271
w(n,n) ..... A003462
w(n,n-1) ... A014915
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Digital Library of Mathematical Functions, Hypergeometric function, analytic properties.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52(3) (2014), 195-202.
FORMULA
From Peter Bala, Jul 16 2013: (Start)
T(n,k) = Sum_{i = 0..k} 3^(k-i)*binomial(n-i,k-i).
O.g.f.: 1/((1 - x*t)*(1 - (1 + 3*x)*t)) = 1 + (1 + 4*x)*t + (1 + 7*x + 13*x^2)*t^2 + ....
The n-th row polynomial is R(n,x) = (1/(2*x + 1))*((3*x + 1)^(n+1) - x^(n+1)). (End)
T(n,k) = T(n-1,k) + 4*T(n-1,k-1) - T(n-2,k-1) - 3*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014
T(n,k) = 3^k * C(n,k) * hyp2F1(1, -k, -n, 1/3) with or without the additional term -0^(n-k)/2 depending on the exact definition of the hypergeometric function used. Compare formulas 15.2.5 and 15.2.6 in the DLMF reference. - Peter Luschny, Jul 23 2014
EXAMPLE
First six rows, for 0 <= k <= n and 0 <= n <= 5:
1
1...4
1...7....13
1...10...34....40
1...13...64....142...121
1...16...103...334...547...364
MAPLE
fission := proc(p, q, n) local d, k;
p(n+1, 0)*q(n, x)+add(coeff(p(n+1, x), x^k)*q(n-k, x), k=1..n);
seq(coeff(%, x, n-k), k=0..n) end:
A193842_row := n -> fission((n, x) -> (x+1)^n, (n, x) -> (x+2)^n, n);
for n from 0 to 5 do A193842_row(n) od; # Peter Luschny, Jul 23 2014
# Alternatively:
p := (n, x) -> add(x^k*(1+3*x)^(n-k), k=0..n): for n from 0 to 7 do [n], PolynomialTools:-CoefficientList(p(n, x), x) od; # Peter Luschny, Jun 18 2017
MATHEMATICA
(* First program *)
z = 10;
p[n_, x_] := (x + 1)^n;
q[n_, x_] := (x + 2)^n
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A193842 *)
TableForm[Table[h[n], {n, 0, z}]] (* A193843 *)
Flatten[Table[h[n], {n, -1, z}]]
(* Second program *)
Table[SeriesCoefficient[((x+3)^(n+1) -1)/(x+2), {x, 0, n-k}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)
PROG
(Sage)
from mpmath import mp, hyp2f1
mp.dps = 100; mp.pretty = True
def T(n, k):
return 3^k*binomial(n, k)*hyp2f1(1, -k, -n, 1/3)-0^(n-k)//2
for n in range(7):
print([int(T(n, k)) for k in (0..n)]) # Peter Luschny, Jul 23 2014
(Sage) # Second program using the 'fission' operation.
def fission(p, q, n):
F = p(n+1, 0)*q(n, x)+add(expand(p(n+1, x)).coefficient(x, k)*q(n-k, x) for k in (1..n))
return [expand(F).coefficient(x, n-k) for k in (0..n)]
A193842_row = lambda k: fission(lambda n, x: (x+1)^n, lambda n, x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 23 2014
(PARI) T(n, k) = sum(j=0, k, 3^(k-j)*binomial(n-j, k-j)); \\ G. C. Greubel, Feb 18 2020
(Magma) [ (&+[3^(k-j)*Binomial(n-j, k-j): j in [0..k]]): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020
CROSSREFS
Cf. A193722 (fusion of P by Q), A193649 (Q-residue), A193843 (mirror of A193842).
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 07 2011
EXTENSIONS
Name and Comments edited by Petros Hadjicostas, Jun 05 2020
STATUS
approved
A090040
(3*6^n + 2^n)/4.
+10
5
1, 5, 28, 164, 976, 5840, 35008, 209984, 1259776, 7558400, 45349888, 272098304, 1632587776, 9795522560, 58773127168, 352638746624, 2115832446976, 12694994616320, 76169967566848, 457019805138944, 2742118830309376
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A090040 is the Q-residue of the triangle A175840, where Q is the triangular array (t(i,j)) given by t(i,j)=1; see A193649 for the definition of Q-residue. - Clark Kimberling, Aug 07 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-12).
FORMULA
G.f.: (1-3*x)/((1-2*x)*(1-6*x)).
E.g.f.: (3*exp(6*x)+exp(2*x))/4 = exp(4*x)*(cosh(2*x)+sinh(2*x)/2).
a(n) = 8*a(n-1) -12*a(n-2), a(0)=1, a(1)=5.
a(n) = (3*6^n+2^n)/4.
a(n)=6*a(n-1)-2^(n-1). - Paul Curtz, Jan 09 2009
Fourth binomial transform of (1, 1, 4, 4, 16, 16, ...). a(n)=sum{k=1..floor(n/2), C(n, 2k)4^(n-k-1)}. - Paul Barry, Nov 22 2003
a(n) = A019590 (mod 4), proof via a(n)=8*a(n-1)-12*a(n-2). - R. J. Mathar, Feb 25 2009
a(n) = Sum_{k, 0<=k<=n} A117317(n,k)*3^k. - Philippe Deléham, Jan 28 2012
MATHEMATICA
LinearRecurrence[{8, -12}, {1, 5}, 30] (* Harvey P. Dale, Nov 23 2014 *)
PROG
(Magma) [(3*6^n+2^n)/4: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011
(PARI) a(n)=(3*6^n+2^n)/4 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Cf. A081335.
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 20 2003
STATUS
approved
A193651
a(n) = ((2*n + 1)!! + 1)/2.
+10
4
1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, 6874655288, 158117071613, 3952926790313, 106729023338438, 3095141676814688, 95949391981255313, 3166329935381425313, 110821547738349885938, 4100397266318945779688, 159915493386438885407813
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Previous name was: Q-residue of the triangle A130534, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
a(404) has 1002 decimal digits. - Michael De Vlieger, Apr 25 2016
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..403
FORMULA
From Peter Luschny, Aug 20 2014: (Start)
a(n) = (2^n*Gamma(n+3/2))/sqrt(Pi) + 1/2.
a(n) = 2^n*Pochhammer(1/2, n+1) + 1/2.
a(n) = ((2*a(n-1) - 2*a(n-2))*n^2 + a(n-2)*n - a(n-1))/(n-1) for n>1, a(0)=1, a(1)=2. (End)
(-n+1)*a(n) +(2*n^2-1)*a(n-1) -n*(2*n-1)*a(n-2)=0. - R. J. Mathar, Feb 19 2015
E.g.f.: (exp(x) + 1/(1-2*x)^(3/2))/2. - Vladimir Reshetnikov, Apr 25 2016
MAPLE
seq((1+doublefactorial(2*n+1))/2, n=0..18); # Peter Luschny, Aug 20 2014
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
u[0, x_] := 1; u[n_, x_] := (x + n)*u[n - 1, x]
p[n_, k_] := Coefficient[u[n, x], x, k]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}] (* A193651 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 6}, {k, 0, n}]] (* A130534 *)
Table[((2 n + 1)!! + 1)/2, {n, 0, 18}] (* or *)
Table[(2^n Gamma[n + 3/2])/Sqrt[Pi] + 1/2, {n, 0, 18}] (* or *)
Table[2^n Pochhammer[1/2, n + 1] + 1/2, {n, 0, 18}] (* Michael De Vlieger, Apr 25 2016 *)
PROG
(Sage)
def A():
n, a, b = 1, 1, 2
yield a
while True:
yield b
n += 1
a, b = b, ((2*(b-a)*n + a)*n - b)/(n-1)
A193651 = A()
[next(A193651) for i in range(19)] # Peter Luschny, Aug 20 2014
CROSSREFS
Cf. A001147, A193649, A130534.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 02 2011
EXTENSIONS
New name from Peter Luschny, Aug 20 2014
STATUS
approved
A193657
First difference of A002627.
+10
4
1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013
Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - John M. Campbell, Apr 20 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From N. J. A. Sloane, Feb 03 2013
FORMULA
E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012
a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012
a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014
a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014
From Peter Bala, Feb 10 2020: (Start)
a(n) = n*A002627(n) + 1.
a(n) = A114870(n) + n!.
a(n) = A296964(n+1) - A296964(n) for n >= 2.
a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
Second column of A176305. (End)
MAPLE
a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1, 1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014
MATHEMATICA
q[n_, k_] := n + k + 1; (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}] (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2, {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Nov 20 2012 *)
PROG
(PARI) a(n) = { sum(k=0, n, if (k <= n-2, binomial(n, k)*(k+1)!, binomial(n, k)^2*k!)); } \\ Michel Marcus, Feb 07 2013
(Sage)
def A193657():
a = 2; b = 7; c = 31; n = 3
yield 1
while True:
yield a
n += 1
a, b, c = b, c, ((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014
CROSSREFS
Cf. A193649, A094727, A082425, A114870, A176305, A296964.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 02 2011
EXTENSIONS
Simpler definition by Peter Luschny, May 30 2014
STATUS
approved
A193664
Q-residue of A049310 (triangle of coefficients of Fibonacci polynomials), where Q=Pascal's triangle. (See Comments.)
+10
3
0, 1, 1, 6, 11, 68, 177, 1215, 4059, 30733, 124408, 1027972, 4862600, 43450761, 234283662, 2247091674, 13563976285, 138780931929, 925063455844, 10044476018973, 73144254450840, 839146997933059, 6618306039456419
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The definition of Q-residue is given at A193649.
LINKS
Table of n, a(n) for n=0..22.
MATHEMATICA
f[n_, x_] := Fibonacci[n, x];
q[n_, k_] := Coefficient[(x + 1)^n, x, k]; (* Pascal's triangle *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
p[n_, k_] := Coefficient[f[n, x], x, k];
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 22}] (* A193664 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* A000110 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
CROSSREFS
Cf. A049310, A193649, A000110.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 02 2011
STATUS
approved
A193650
Q-residue of the coefficient triangle (A076699) of the cyclotomic polynomials, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)
+10
2
1, 0, 3, 7, 10, 31, 24, 127, 136, 292, 352, 2047, 1664, 8191, 5504, 9664, 32896, 131071, 116736, 524287, 419840, 603904, 1398784, 8388607, 7897088, 17318416
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
For the definition of Q-residue, see A193649.
LINKS
Table of n, a(n) for n=0..25.
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Coefficient[Cyclotomic[n, x], x, k]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 25}] (* A193650 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 8}, {k, 0, n}]] (* A076699 *)
CROSSREFS
Cf. A193649, A076699.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 02 2011
STATUS
approved
A193653
Q-residue of the Delannoy triangle A008288, where Q is the triangular array (t(i,j)) given by t(i,j)=1.
+10
2
1, 2, 6, 20, 70, 248, 882, 3140, 11182, 39824, 141834, 505148, 1799110, 6407624, 22821090, 81278516, 289477726, 1030990208, 3671926074, 13077758636, 46577128054, 165886901432, 590814960402, 2104218684068, 7494285973006, 26691295287152, 95062457807466
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
For the definition of Q-residue, see A193649.
This sequence gives the number of closed walks from the two vertices having loops in the digraph defined by its adjacency matrix A = (2,1,1; 1,2,1; 1,1,0). - David Neil McGrath, Aug 22 2014
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).
FORMULA
From David Neil McGrath, Aug 22 2014: (Start)
a(n) = 4*a(n-1) - a(n-2) - 2*a(n-3).
a(n-1) = (1,1) and (2,2) elements of A^(n-1) where A=(2,1,1; 1,2,1; 1,1,0) and n>1. (End)
G.f.: (1-2*t-t^2)/(1-4*t+t^2+2*t^3). - Robert Israel, Aug 22 2014
a(n) = (34 + (17-3*sqrt(17))*((3-sqrt(17))/2)^n + ((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68. - Colin Barker, Sep 02 2016
From G. C. Greubel, May 25 2021: (Start)
a(n) = (1/2)*(1 + (i*sqrt(2))^n * ChebyshevU(n, -3*i/(2*sqrt(2)))).
a(n) = (1/2)*( 1 + Sum_{j=0..floor(n/2)} binomial(n-k,k)*2^k*3^(n-2*k) ). (End)
MATHEMATICA
(* First program *)
q[n_, k_] := 1;
r[0] = 1; r[k_]:= Sum[q[k-1, i]*r[k-1-i], {i, 0, k-1}]
p[n_, k_]:= p[n, k]= If[k==0 || k==n, 1, p[n-1, k-1] + p[n-2, k-1] + p[n-1, k]]; (* A008288, Delannoy *)
v[n_]:= Sum[p[n, k]*r[n-k], {k, 0, n}];
Table[v[n], {n, 0, 16}] (* A193653 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
(* Second program *)
LinearRecurrence[{4, -1, -2}, {1, 2, 6}, 40] (* G. C. Greubel, May 25 2021 *)
PROG
(PARI) Vec((1-2*t-t^2)/(1-4*t+t^2+2*t^3) + O(t^40)) \\ Michel Marcus, Aug 23 2014
(PARI) a(n) = round((34+(17-3*sqrt(17))*((3-sqrt(17))/2)^n+((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68) \\ Colin Barker, Sep 02 2016
(Magma) [n le 3 select Factorial(n) else 4*Self(n-1) -Self(n-2) -2*Self(n-3): n in [1..41]]; // G. C. Greubel, May 25 2021
(Sage) [(1/2)*(1 + sum(binomial(n-k, k)*2^k*3^(n-2*k) for k in (0..n//2))) for n in (0..40)] # G. C. Greubel, May 25 2021
CROSSREFS
Cf. A008288, A193649.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 02 2011
STATUS
approved
A193654
Q-residue of the triangle p(n,k)=floor((n+1)/(n+k+2)/2), 0<=k<=n, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)
+10
2
1, 7, 28, 94, 275, 765, 2002, 5116, 12625, 30715, 73040, 172026, 398671, 917497, 2086222, 4718584, 10573133, 23592951, 52254028
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
For the definition of Q-residue, see A193649.
LINKS
Table of n, a(n) for n=0..18.
FORMULA
Conjecture: G.f.: ( -1-2*x+4*x^2+4*x^3-8*x^5 ) / ( (1+x)*(2*x+1)*(x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Feb 19 2015
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Floor[(n + 1) (n + k + 2)/2]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 16}] (* A193654 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}] (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
CROSSREFS
Cf. A193649, A193655.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 02 2011
STATUS
approved
A193655
Q-residue of the triangle p(n,k)=floor(1/2+(n+1)/(n+k+2)/2), 0<=k<=n, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)
+10
2
1, 7, 29, 94, 280, 765, 2023, 5116, 12710, 30715, 73381, 172026, 400036, 917497, 2091683, 4718584, 10594978, 23592951, 52341409, 115343350, 253405856
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
For the definition of Q-residue, see A193649.
LINKS
Table of n, a(n) for n=0..20.
FORMULA
Conjecture: G.f.: ( -1-2*x+3*x^2+9*x^3-8*x^4-4*x^5 ) / ( (1+x)*(2*x+1)*(x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Feb 19 2015
MATHEMATICA
q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Floor[1/2 + (n + 1) (n + k + 2)/2]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}] (* A193655 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
CROSSREFS
Cf. A193649, A193654.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Aug 02 2011
STATUS
approved
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