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Search: a026780 -id:a026780
Displaying 1-10 of 31 results found.
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A026781
a(n) = T(2n,n), T given by A026780.
+20
14
1, 3, 12, 53, 246, 1178, 5768, 28731, 145108, 741392, 3825418, 19907156, 104370554, 550816506, 2924018194, 15603778253, 83661779470, 450479003038, 2435009205992, 13208558795146, 71879906857596, 392320357251928, 2147102400154768, 11780181236675858, 64782405317073968, 357022158144941548
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of paths from (0,0) to (n,n) in the directed graph having vertices (i,j) and edges (i,j)-to-(i+1,j) and (i,j)-to-(i,j+1) for i,j>=0 and edges (i,i+h)-to-(i+1,i+h+1) for i>=0, h>=0.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
M. A. Alekseyev. On Enumeration of Dyck-Schroeder Paths. Journal of Combinatorial Mathematics and Combinatorial Computing 106 (2018), 59-68; arXiv:1601.06158 [math.CO], 2016-2018.
FORMULA
O.g.f.: S(x)/(1-x*C(x)*S(x)) = (S(x)-C(x))/(x*C(x)), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - Max Alekseyev, Jan 13 2015
D-finite with recurrence 2*n*(132*n-445)*(n+2)*(n+1)*a(n) -n*(n+1) *(5587*n^2 -23082*n +12800)*a(n-1) +2*n*(n-1)*(22870*n^2 -114505*n +116854)*a(n-2) +2*(-90081*n^4 +818062*n^3 -2626791*n^2 +3517598*n -1622544)*a(n-3) +4*(85519*n^4 -1071535*n^3 +4986308*n^2 -10177616*n +7647024)*a(n-4) +(-269235*n^4 +4490125*n^3 -27985152*n^2 +77217236*n -79534224)*a(n-5) +4*(2*n-11)*(8203*n^3 -117312*n^2 +557264*n -879984)*a(n-6) -4*(n-6)*(307*n -1414) *(2*n-11) *(2*n-13)*a(n-7)=0. - R. J. Mathar, Feb 20 2020
MAPLE
seq(coeff(series(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2))), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
CoefficientList[Series[2*(1-x -Sqrt[1-6*x+x^2])/(4*x -(1 -Sqrt[1-4*x])*(1 -x -Sqrt[1-6*x+x^2])), {x, 0, 30}], x] (* G. C. Greubel, Nov 02 2019 *)
PROG
(PARI) C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51) ))/2/x; Vec(S/(1-x*C*S)) /* Max Alekseyev, Jan 13 2015 */
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2*(1-x -Sqrt(1-6*x+x^2))/(4*x -(1 -Sqrt(1-4*x))*(1 -x -Sqrt(1-6*x+x^2))) )); // G. C. Greubel, Nov 02 2019
(Sage)
def A026781_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(2*(1-x -sqrt(1-6*x+x^2))/(4*x -(1 -sqrt(1-4*x))*(1 -x -sqrt(1-6*x+x^2)))).list()
A026781_list(30) # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026671.
Cf. A026780, A026782, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
EXTENSIONS
More terms from Max Alekseyev, Jan 13 2015
STATUS
approved
A026787
a(n) = Sum_{k=0..n} T(n,k), T given by A026780.
+20
12
1, 2, 5, 11, 26, 58, 136, 306, 717, 1625, 3813, 8697, 20451, 46909, 110563, 254855, 602042, 1393746, 3299304, 7666786, 18182976, 42391546, 100704606, 235452416, 560147414, 1312916040, 3127406812, 7346213746, 17518138314, 41228281888, 98408997716, 231990850378, 554207752781, 1308436686305, 3128033585157
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
O.g.f.: F(x^2)*(1/(1-x*S(x^2))+C(x^2)*x/(1-x*C(x^2))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781. - Max Alekseyev, Jan 13 2015
C(x^2)/(1-x*C(x^2)) above is the o.g.f. for A001405. 1/(1-x*S(x^2)) above is the o.g.f for A026003 starting with an additional 1: 1,1,1,3,5,13,25,... - R. J. Mathar, Feb 10 2022
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq( add(T(n, k), k=0..n), n=0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
EXTENSIONS
More terms from Max Alekseyev, Jan 13 2015
STATUS
approved
A026773
a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.
+20
11
1, 4, 17, 76, 352, 1674, 8129, 40156, 201236, 1020922, 5234660, 27089726, 141335846, 742712598, 3927908193, 20891799036, 111688381228, 599841215226, 3234957053984, 17512055200470, 95125188934942, 518340392855286, 2832580291316092, 15520177744727766
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
From Vladeta Jovovic, Nov 23 2003: (Start)
a(n) = A006318(n) - A000108(n).
G.f.: (sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2. (End)
From Paul Barry, May 19 2005: (Start)
a(n) = Sum_{k=0..n} C(n+k+1, n+1)*C(n+1, k)/(k+1).
a(n) = Sum_{k=0..n+1} C(n+2, k)*C(n+k, n+1)/(n+2). (End)
D-finite with recurrence n*(n+1)*a(n) -n*(11*n-7)*a(n-1) +(37*n^2-95*n+54)*a(n-2) +(-49*n^2+269*n-354)*a(n-3) +6*(9*n^2-71*n+138)*a(n-4) -4*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Aug 05 2021
MAPLE
seq(coeff(series((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2, x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019
MATHEMATICA
Rest@CoefficientList[Series[(Sqrt[1-4*x] - Sqrt[1-6*x+x^2])/(2*x) -1/2, {x, 0, 30}], x] (* G. C. Greubel, Nov 01 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2) \\ G. C. Greubel, Nov 01 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-4*x) - Sqrt(1-6*x+x^2))/(2*x) -1/2 )); // G. C. Greubel, Nov 01 2019
(Sage)
def A026773_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2).list()
a=A026773_list(30); a[1:] # G. C. Greubel, Nov 01 2019
(GAP) List([0..30], n-> Sum([0..n], k-> Binomial(n+1, k)*Binomial(n+k+1, n+1)/(k+1) )); # G. C. Greubel, Nov 01 2019
CROSSREFS
Cf. A026769, A026770, A026771, A026772, A026774, A026775, A026776, A026777, A026778, A026779.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026782
a(n) = T(2n,n-1), T given by A026780.
+20
11
1, 7, 40, 217, 1158, 6150, 32656, 173719, 926664, 4958556, 26619438, 143365880, 774562478, 4197344582, 22810572062, 124300860689, 679081142350, 3718894341450, 20412141531664, 112276061739814, 618806031336236, 3416954495002676
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq(T(2*n, n-1), n=1..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[T(2*n, n-1) for n in (1..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026783, A026784, A026785, A026786, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026783
a(n) = T(2n, n-2), T given by A026780.
+20
11
1, 11, 84, 557, 3446, 20514, 119336, 684227, 3886460, 21939528, 123347842, 691644044, 3871738018, 21652138770, 121026492186, 676391629701, 3780636102222, 21137831159462, 118234019051048, 661686074145618, 3705252204960252
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..500
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq(T(2*n, n-2), n=2..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n, n-2], {n, 2, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[T(2*n, n-2) for n in (2..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026784, A026785, A026786, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026784
a(n) = T(2n-1, n-1), T given by A026780.
+20
11
1, 5, 24, 117, 580, 2916, 14834, 76221, 395048, 2063104, 10847078, 57373672, 305110106, 1630489090, 8751851866, 47166202181, 255128842340, 1384688987728, 7538592535170, 41159292861980, 225315261459390, 1236441650047554
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq(T(2*n-1, n-1), n=1..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n-1, n-1], {n, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[T(2*n-1, n-1) for n in (1..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026783, A026785, A026786, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026785
a(n) = T(2n-1, n-2), T given by A026780.
+20
11
1, 9, 60, 361, 2076, 11672, 64842, 357897, 1968788, 10813804, 59372770, 326086492, 1792293014, 9861375614, 54324086446, 299651439321, 1655124211372, 9154654655044, 50704627346170, 281214708137032, 1561706813618886
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 2..500
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq(T(2*n-1, n-2), n=2..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[2*n-1, n-2], {n, 2, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[T(2*n-1, n-2) for n in (2..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026783, A026784, A026786, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026786
a(n) = T(n, floor(n/2)), T given by A026780.
+20
11
1, 1, 3, 5, 12, 24, 53, 117, 246, 580, 1178, 2916, 5768, 14834, 28731, 76221, 145108, 395048, 741392, 2063104, 3825418, 10847078, 19907156, 57373672, 104370554, 305110106, 550816506, 1630489090, 2924018194, 8751851866
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq(T(n, floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[T[n, Floor[n/2]], {n, 0, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[T(n, floor(n/2)) for n in (0..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026787, A026788, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026788
a(n) = Sum_{k=0..floor(n/2)} T(n,k), T given by A026780.
+20
11
1, 1, 4, 6, 20, 34, 105, 191, 563, 1071, 3057, 6007, 16745, 33729, 92332, 189662, 511812, 1068178, 2849404, 6025594, 15921514, 34043204, 89242582, 192621212, 501574732, 1091400122, 2825710822, 6192005260, 15952433940, 35172854946
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq( add(T(n, k), k=0..floor(n/2)), n=0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[sum(T(n, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026787, A026789, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A026789
a(n) = Sum_{i=0..n} Sum_{j=0..n} T(i,j), T given by A026780.
+20
11
1, 3, 8, 19, 45, 103, 239, 545, 1262, 2887, 6700, 15397, 35848, 82757, 193320, 448175, 1050217, 2443963, 5743267, 13410053, 31593029, 73984575, 174689181, 410141597, 970289011, 2283205051, 5410611863, 12756825609, 30274963923
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
MAPLE
T:= proc(n, k) option remember;
if n<0 then 0;
elif k=0 or k =n then 1;
elif k <= n/2 then
procname(n-1, k-1)+procname(n-2, k-1)+procname(n-1, k) ;
else
procname(n-1, k-1)+procname(n-1, k) ;
fi ;
end proc:
seq( add(add(T(j, k), k=0..n), j=0..n), n=0..30); # G. C. Greubel, Nov 02 2019
MATHEMATICA
T[n_, k_]:= T[n, k]= If[n<0, 0, If[k==0 || k==n, 1, If[k<=n/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k] ]]];
Table[Sum[T[j, k], {k, 0, n}, {j, 0, n}], {n, 0, 30}] (* G. C. Greubel, Nov 02 2019 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n<0): return 0
elif (k==0 or k==n): return 1
elif (k<=n/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
else: return T(n-1, k-1) + T(n-1, k)
[sum( sum( T(j, k) for k in (0..n)) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 02 2019
CROSSREFS
Partial sums of A026787.
Cf. A026780, A026781, A026782, A026783, A026784, A026785, A026786, A026788, A026790.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
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