A387010 a(n) = Sum_{k=0..n} binomial(4*n+3,k).

1, 8, 67, 576, 5036, 44552, 397594, 3572224, 32267668, 292750368, 2665685155, 24347665728, 222972599812, 2046626681072, 18823260696452, 173427623923712, 1600383346290116, 14789063407109600, 136838247669241276, 1267571539176770816, 11754134090271100336

Offset: 0

Author

Seiichi Manyama, Aug 12 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A066381, A386811, A387009, A387011.

Cf. A002293.

Programs

Magma

[&+[Binomial(4*n+3, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 16 2025

Mathematica

Table[Sum[Binomial[4*n+3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 16 2025 *)

PARI

a(n) = sum(k=0, n, binomial(4*n+3, k));

Formula

a(n) = [x^n] (1+x)^(4*n+3)/(1-x).

a(n) = [x^n] 1/((1-x)^(3*n+3) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+3,k) * binomial(4*n-k+2,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k+2,n-k).

G.f.: g^4/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.

D-finite with recurrence: 128*(4*n-1)*(2*n-1)*(4*n-3)*(11*n^2+8*n+1)*a(n-2) -8*(946*n^5-434*n^4-518*n^3+143*n^2+46*n-3)*a(n-1) +3*n*(3*n+1)*(3*n+2)*(11*n^2-14*n+4)*a(n) = 0. - Georg Fischer, Aug 17 2025