Displaying 31-40 of 72 results found.
Triangle read by rows: T(n,k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w + 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
+10
4
1, 2, 2, 4, 6, 4, 8, 16, 16, 8, 16, 40, 52, 40, 16, 32, 96, 152, 152, 96, 32, 64, 224, 416, 504, 416, 224, 64, 128, 512, 1088, 1536, 1536, 1088, 512, 128, 256, 1152, 2752, 4416, 5136, 4416, 2752, 1152, 256, 512, 2560, 6784, 12160, 16032, 16032, 12160, 6784, 2560, 512
COMMENTS
Pascal-like triangle: start with 1 at top; every subsequent entry is the sum of everything above you, plus 1.
FORMULA
G.f.: 1/(1 - 2*z - 2*w + 2*z*w).
T(n, k) = Sum_{j=0..n} (-1)^j*2^(n + k - j)*C(n, j)*C(n + k - j, n).
T(n, k) = 2^n*binomial(n, k)*hypergeom([-k, k - n], [-n], 1/2). - Peter Luschny, Nov 26 2021 T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).
EXAMPLE
Triangle begins as:
n\k [0] [1] [2] [3] [4] [5] [6] ...
[0] 1;
[1] 2, 2;
[2] 4, 6, 4;
[3] 8, 16, 16, 8;
[4] 16, 40, 52, 40, 16;
[5] 32, 96, 152, 152, 96, 32;
[6] 64, 224, 416, 504, 416, 224, 64;
...
MAPLE
read transforms; SERIES2(1/(1-2*z-2*w+2*z*w), x, y, 12): SERIES2TOLIST(%, x, y, 12);
# Alternative
T := (n, k) -> 2^n*binomial(n, k)*hypergeom([-k, -n + k], [-n], 1/2):
for n from 0 to 10 do seq(simplify(T(n, k)), k = 0 .. n) end do; # Peter Luschny, Nov 26 2021
MATHEMATICA
Table[(-1)^k*2^n*JacobiP[k, -n-1, 0, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Oct 04 2017; May 21 2023 *)
PROG
(Magma)
A059474:= func< n, k | (&+[(-1)^j*2^(n-j)*Binomial(n-k, j)*Binomial(n-j, n-k): j in [0..n-k]]) >;
(SageMath)
def A059474(n, k): return 2^n*binomial(n, k)*simplify(hypergeometric([-k, k-n], [-n], 1/2))
CROSSREFS
See A059576 for a similar triangle.
a(0)=0; a(1)=1; a(n) = a(n-1) + (3 + (-1)^n)*a(n-2)/2.
+10
4
0, 1, 1, 2, 4, 6, 14, 20, 48, 68, 164, 232, 560, 792, 1912, 2704, 6528, 9232, 22288, 31520, 76096, 107616, 259808, 367424, 887040, 1254464, 3028544, 4283008, 10340096, 14623104, 35303296, 49926400, 120532992, 170459392, 411525376
FORMULA
G.f.: x*(1+x-2*x^2)/(1-4*x^2+2*x^4).
a(n) = 4*a(n-2) - 2*a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=2. - Harvey P. Dale, May 24 2013
EXAMPLE
a(4) = a(3) + 2*a(2) = 2 + 2 = 4.
MATHEMATICA
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]+(3+(-1)^n) (a[n-2])/2}, a, {n, 40}] (* or *) LinearRecurrence[{0, 4, 0, -2}, {0, 1, 1, 2}, 40] (* Harvey P. Dale, May 24 2013 *)
PROG
(PARI) { for (n=0, 200, if (n>1, a=a1 + (3 + (-1)^n)*a2/2; a2=a1; a1=a, if (n==0, a=a2=0, a=a1=1)); write("b062112.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 01 2009 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x-2*x^2)/(1-4*x^2+2*x^4))); // G. C. Greubel, Oct 16 2018
Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
+10
4
1, 4, 1, 13, 8, 1, 40, 42, 12, 1, 121, 184, 87, 16, 1, 364, 731, 496, 148, 20, 1, 1093, 2736, 2454, 1040, 225, 24, 1, 3280, 9844, 11064, 6170, 1880, 318, 28, 1, 9841, 34448, 46738, 32624, 13015, 3080, 427, 32, 1, 29524, 118101, 188208, 158724, 79044, 24381, 4704
COMMENTS
Subtriangle of triangle given by (0, 4, -3/4, 3/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012
FORMULA
Riordan array (1/(1-4x+3x^2), x/(1-4x+3x^2)); number triangle T(n,k) = Sum_{j=0..n} binomial(n-j,k)*binomial(k+j,j)*3^j.
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(2,1) = 4, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Oct 31 2013 O.g.f.: 1/(1 - 4*x + 3*x^2 - x*y) = 1 + (4 + y)*x + (13 + 8*y + y^2)*x^2 + ....
Recurrence for row polynomials: R(n,y) = (4 + y)*R(n-1,y) - 3*R(n-2,y) with R(0,y) = 1 and R(1,y) = 4 + y.
The row reverse polynomial y^n*R(n,1/y) is equal to the numerator polynomial of the finite continued fraction 1 + y/(1 + 3*y/(1 + ... + y/(1 + 3*y/(1)))) (with 2*n partial numerators). Cf. A110441. (End)
EXAMPLE
Triangle begins
1;
4, 1;
13, 8, 1;
40, 42, 12, 1;
121, 184, 87, 16, 1;
364, 731, 496, 148, 20, 1;
Triangle T(n,k), 0 <= k <= n, given by (0, 4, -3/4, 3/4, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, ...) begins:
1;
0, 1;
0, 4, 1;
0, 13, 8, 1;
0, 40, 42, 12, 1;
0, 121, 184, 87, 16, 1;
0, 364, 731, 496, 148, 20, 1;
MATHEMATICA
With[{n = 10}, DeleteCases[#, 0] & /@ Rest@ CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 4 x + 3 x^2 - x y), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)
Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.
+10
4
1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1
COMMENTS
The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.
FORMULA
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.
T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
Sum_{k=0..n} T(n, k) = A001906(n+1). Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1). Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n). Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n). T(n, n) = (-1)^n.
T(n, n-2) = (-1)^n* A139278(n-1), n >= 2.
EXAMPLE
Triangle begins as:
1;
4, -1;
15, -8, 1;
56, -46, 12, -1;
209, -232, 93, -16, 1;
780, -1091, 592, -156, 20, -1;
2911, -4912, 3366, -1200, 235, -24, 1;
10864, -21468, 17784, -8010, 2120, -330, 28, -1;
MAPLE
A123966x := proc(n, x)
local A, r, c ;
A := Matrix(1..n, 1..n) ;
for r from 1 to n do
for c from 1 to n do
A[r, c] :=0 ;
if r = c then
A[r, c] := A[r, c]+4 ;
elif abs(r-c)= 1 then
A[r, c] := A[r, c]-1 ;
end if;
end do:
end do:
(-1)^n*LinearAlgebra[CharacteristicPolynomial](A, x) ;
end proc;
coeftayl( A123966x(n, x), x=0, k) ;
end proc:
MATHEMATICA
(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n, d}, {m, d}];
Table[M[d], {d, 10}]
Table[Det[M[d]], {d, 10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d, 10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 10}]]//Flatten
(* Recursive Polynomial form*)
p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n, 0, 12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)
PROG
(Magma)
m:=12;
R<x>:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n, k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
(SageMath)
def A124029(n, k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
a(n) = ((9 + sqrt(2))^n - (9 - sqrt(2))^n)/(2*sqrt(2)).
+10
4
1, 18, 245, 2988, 34429, 383670, 4186169, 45041112, 480032665, 5082340122, 53559541661, 562566880260, 5895000053461, 61667217421758, 644304909368225, 6725778192309168, 70163919621475249, 731614075994130210
COMMENTS
Eighth binomial transform of A048697. lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(2) = 10.4142135623....
FORMULA
a(n) = 18*a(n-1) - 79*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009 G.f.: x/(1 - 18*x + 79*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009 a(n) = Sum[Binomial[n - 1 - i, i] (-1)^i * 18^(n - 1 - 2 i) * 79^i, {i, 0, Floor[(n - 1)/2]}]. - Sergio Falcon, Sep 21 2009
MATHEMATICA
LinearRecurrence[{18, -79}, {1, 18}, 25] (* G. C. Greubel, Aug 22 2016 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
a(n) = ((5 + 2*sqrt(2))*(2 + sqrt(2))^n + (5 - 2*sqrt(2))*(2 - sqrt(2))^n)/2.
+10
4
5, 14, 46, 156, 532, 1816, 6200, 21168, 72272, 246752, 842464, 2876352, 9820480, 33529216, 114475904, 390845184, 1334428928, 4556025344, 15555243520, 53108923392, 181325206528, 619082979328, 2113681504256, 7216560058368
FORMULA
a(n) = 4*a(n-1) - 2*a(n-2) for n > 1; a(0) = 5, a(1) = 14.
G.f.: (5-6*x)/(1-4*x+2*x^2).
E.g.f.: exp(2*x)*( 5*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017
MATHEMATICA
LinearRecurrence[{4, -2}, {5, 14}, 30] (* Harvey P. Dale, Jan 31 2017 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-2); S:=[ ((5+2*r)*(2+r)^n+(5-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 06 2009 (PARI) x='x+O('x^50); Vec((5-6*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jul 29 2017
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009
Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.
+10
4
1, 3, 2, 6, 9, 4, 10, 25, 24, 8, 15, 55, 85, 60, 16, 21, 105, 231, 258, 144, 32, 28, 182, 532, 833, 728, 336, 64, 36, 294, 1092, 2241, 2720, 1952, 768, 128, 45, 450, 2058, 5301, 8361, 8280, 5040, 1728, 256, 55, 660, 3630, 11385, 22363, 28610, 23920
COMMENTS
Column 1: triangular numbers, A000217Coefficient of v(n,x): 2^(n-1)
Alternating row sums: 1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510. Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - Tom Copeland, Jan 09 2017
FORMULA
u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
3....2
6....9....4
10...25...24...8
15...55...85...60...16
First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *) Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)
Power floor-ceiling sequence of 2+sqrt(2).
+10
4
3, 11, 37, 127, 433, 1479, 5049, 17239, 58857, 200951, 686089, 2342455, 7997641, 27305655, 93227337, 318298039, 1086737481, 3710353847, 12667940425, 43251054007, 147668335177, 504171232695, 1721348260425, 5877050576311, 20065505784393, 68507921984951
COMMENTS
See A214992 for a discussion of power floor-ceiling sequence and power floor-ceiling function, p2(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p2(r) = (11 + 8*sqrt(2))/7. a(n) is the number of ways to tile a 2 X (n+1) strip, with one extra square at the top left corner, using 1 X 1 squares, 2 X 2 squares, and 1 X 2 dominoes (either horizontal or vertical). This picture shows a(1) = 11.
_ _ _ _ _ _ _ _ _ _ _
|_|_ |_|_ | |_ |_|_ |_|_ |_|_ |_|_ | |_ | |_ |_|_ |_|_
|_|_| | | |_|_| | |_| |_| | |___| |_|_| |_|_| |_| | |___| | | |
|_|_| |___| |_|_| |_|_| |_|_| |_|_| |___| |___| |_|_| |___| |_|_|
(End)
FORMULA
a(n) = ceiling(x*a(n-1)) if n is odd, a(n) = floor(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = floor(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (3 + 2*x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/7)*((-1)^(1+n) + (11-8*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(11+8*sqrt(2))). - Colin Barker, Nov 13 2017
EXAMPLE
a(0) = floor(r) = 3, where r = 2+sqrt(2).
a(1) = ceiling(3*r) = 11; a(2) = floor(11*r) = 37.
MATHEMATICA
x = 2 + Sqrt[2]; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}] (* A007052 *) Table[p2[n], {n, 0, z}] (* A214996 *) Table[p3[n], {n, 0, z}] (* A214997 *) Table[p4[n], {n, 0, z}] (* A007070 *)
PROG
(PARI) Vec((3 + 2*x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((3+2*x-2*x^2)/(1-3*x-2*x^2+2*x^3))) // G. C. Greubel, Feb 02 2018
Power ceiling-floor sequence of 2+sqrt(2).
+10
4
4, 13, 45, 153, 523, 1785, 6095, 20809, 71047, 242569, 828183, 2827593, 9654007, 32960841, 112535351, 384219721, 1311808183, 4478793289, 15291556791, 52208640585, 178251448759, 608588513865, 2077851157943, 7094227604041, 24221208100279, 82696377193033
COMMENTS
See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(2), and the limit p3(r) = 3.8478612632206289... a(n) is the number of words over {0,1,2,3} of length n+1 that avoid 23, 32, and 33. As an example, a(2)=45 corresponds to the 45 such words of length 3; these are all 64 words except for the 19 prohibited cases which are 320, 321, 322, 323, 230, 231, 232, 233, 330, 331, 332, 333, 023, 123, 223, 032, 132, 033, 133. - Greg Dresden and Mina BH Arsanious, Aug 09 2023
FORMULA
a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1)) if n is even, where x = 2+sqrt(2) and a(0) = ceiling(x).
a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3).
G.f.: (4 + x - 2*x^2)/(1 - 3*x - 2*x^2 + 2*x^3).
a(n) = (1/14)*(2*(-1)^n + (27-19*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(27+19*sqrt(2))). - Colin Barker, Nov 13 2017
EXAMPLE
a(0) = ceiling(r) = 4, where r = 2+sqrt(2);
a(1) = floor(4*r) = 13; a(2) = ceiling(13*r) = 45.
MATHEMATICA
CoefficientList[Series[(4+x-2*x^2)/(1-3*x-2*x^2+2*x^3), {x, 0, 50}], x] (* G. C. Greubel, Feb 01 2018 *)
PROG
(PARI) Vec((4 + x - 2*x^2) / ((1 + x)*(1 - 4*x + 2*x^2)) + O(x^40)) \\ Colin Barker, Nov 13 2017 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((4 +x-2*x^2)/(1-3*x-2*x^2+2*x^3))) // G. C. Greubel, Feb 01 2018
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x + k*x^2).
+10
4
1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0
FORMULA
T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, ...
0, 3, 14, 33, 60, 95, ...
0, 4, 48, 180, 448, 900, ...
0, 5, 164, 981, 3344, 8525, ...
0, 6, 560, 5346, 24960, 80750, ...
MAPLE
T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
MATHEMATICA
T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)
PROG
(PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));
(PARI) T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));
(PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));
CROSSREFS
Main diagonal gives (-1) * A109520(n+1).
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