A026961 - OEIS

%I #17 Jun 23 2024 10:34:58

%S 1,2,11,34,138,492,1830,6804,25576,96728,367932,1405884,5392590,

%T 20751504,80076872,309748096,1200669828,4662772672,18137643524,

%U 70657441212,275620281310,1076429623256,4208562777342,16470788108008,64519534566362,252948764993472,992453764928050

%N Self-convolution of array T given by A026626.

%H G. C. Greubel, <a href="/A026961/b026961.txt">Table of n, a(n) for n = 0..1000</a>

%F From _G. C. Greubel_, Jun 21 2024: (Start)

%F a(n) = Sum_{k=0..n} T(n, k)*T(n, n-k). - _G. C. Greubel_, Jun 21 2024

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

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

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

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

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

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

%t 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]]];

%t A026961[n_]:= A026961[n] = Sum[T[n,k]*T[n,n-k], {k,0,n}];

%t Table[A026961[n], {n,0,50}] (* _G. C. Greubel_, Jun 21 2024 *)

%o (Magma)

%o p1:= func< n | -1864800 + 1239076*n + 7915984*n^2 - 11263411*n^3 + 5406551*n^4 - 1042185*n^5 + 65025*n^6 >;

%o p2:= func< n | -4505760 + 7236856*n + 10545958*n^2 - 20700889*n^3 + 10823147*n^4 - 2188767*n^5 + 143055*n^6 >;

%o p3:= func< n | -1522080 + 2667320*n + 3116288*n^2 - 6715322*n^3 + 3619972*n^4 - 755718*n^5 + 52020*n^6 >;

%o p4:= func< n | 42*(-376320 + 434044*n + 1225808*n^2 - 1997637*n^3 + 1002947*n^4 - 199767*n^5 + 13005*n^6) >;

%o p5:= func< n | 2*(-559440 + 1665230*n - 243157*n^2 - 1361078*n^3 + 898312*n^4 - 195432*n^5 + 13005*n^6) >;

%o 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

%o (SageMath)

%o @CachedFunction

%o def T(n, k): # T = A026626

%o     if (k==0 or k==n): return 1

%o     elif (k==1 or k==n-1): return int(3*n//2)

%o     else: return T(n-1, k-1) + T(n-1, k)

%o def A026961(n): return sum(T(n,k)*T(n,n-k) for k in range(n+1))

%o [A026961(n) for n in range(41)] # _G. C. Greubel_, Jun 21 2024

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

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

%Y Cf. A026965.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E More terms from _Sean A. Irvine_, Oct 20 2019