A026961 - OEIS

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

(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

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.