A026961 Self-convolution of array T given by A026626.

1, 2, 11, 34, 138, 492, 1830, 6804, 25576, 96728, 367932, 1405884, 5392590, 20751504, 80076872, 309748096, 1200669828, 4662772672, 18137643524, 70657441212, 275620281310, 1076429623256, 4208562777342, 16470788108008, 64519534566362, 252948764993472, 992453764928050

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

From G. C. Greubel, Jun 21 2024: (Start)

a(n) = Sum_{k=0..n} T(n, k)*T(n, n-k). - G. C. Greubel, Jun 21 2024

a(n) = (p1(n)*a(n-1) + p2(n)*a(n-2) + p3(n)*a(n-3) + p4(n))/p5(n), where

p1(n) = 22589280 - 75610404*n + 85542748*n^2 - 44611965*n^3 + 11592851*n^4 - 1432335*n^5 + 65025*n^6.

p2(n) = 32659200 - 131052480*n + 161621002*n^2 - 88742247*n^3 + 23912807*n^4 - 3047097*n^5 + 143055*n^6.

p3(n) = 2*(5034960 - 21140910*n + 26659783*n^2 - 14896395*n^3 + 4089431*n^4 - 533919*n^5 + 26010*n^6).

p4(n) = 42*(3628800 - 13099136*n + 15429146*n^2 - 8267195*n^3 + 2196857*n^4 - 277797*n^5 + 13005*n^6).

p5(n) = 2*n*(-6580128 + 11379344*n - 7168746*n^2 + 2070547*n^3 - 273462*n^4 + 13005*n^5). (End)

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, (6*n-1 + (-1)^n)/4, T[n-1, k-1] +T[n-1, k]]];
A026961[n_]:= A026961[n] = Sum[T[n, k]*T[n, n-k], {k, 0, n}];
Table[A026961[n], {n, 0, 50}] (* G. C. Greubel, Jun 21 2024 *)

Prog

(Magma)
p1:= func< n | -1864800 + 1239076*n + 7915984*n^2 - 11263411*n^3 + 5406551*n^4 - 1042185*n^5 + 65025*n^6 >;
p2:= func< n | -4505760 + 7236856*n + 10545958*n^2 - 20700889*n^3 + 10823147*n^4 - 2188767*n^5 + 143055*n^6 >;
p3:= func< n | -1522080 + 2667320*n + 3116288*n^2 - 6715322*n^3 + 3619972*n^4 - 755718*n^5 + 52020*n^6 >;
p4:= func< n | 42*(-376320 + 434044*n + 1225808*n^2 - 1997637*n^3 + 1002947*n^4 - 199767*n^5 + 13005*n^6) >;
p5:= func< n | 2*(-559440 + 1665230*n - 243157*n^2 - 1361078*n^3 + 898312*n^4 - 195432*n^5 + 13005*n^6) >;
I:=[11, 34, 138]; [1, 2] cat [n le 3 select I[n] else (p1(n)*Self(n-1) + p2(n)*Self(n-2) + p3(n)*Self(n-3) + p4(n))/p5(n) : n in [1..40]]; // G. C. Greubel, Jun 21 2024
(SageMath)
@CachedFunction
def T(n, k): # T = A026626
    if (k==0 or k==n): return 1
    elif (k==1 or k==n-1): return int(3*n//2)
    else: return T(n-1, k-1) + T(n-1, k)
def A026961(n): return sum(T(n, k)*T(n, n-k) for k in range(n+1))
[A026961(n) for n in range(41)] # G. C. Greubel, Jun 21 2024

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.

Cf. A026633, A026634, A026635, A026636, A026962, A026963, A026964.

Cf. A026965.

Sequence in context: A328152 A209033 A222650 * A026971 A323683 A071801

Adjacent sequences: A026958 A026959 A026960 * A026962 A026963 A026964

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Sean A. Irvine, Oct 20 2019

Status

approved