A387036 a(n) = Sum_{k=0..n} binomial(4*n-2,k).

1, 3, 22, 176, 1471, 12616, 110056, 971712, 8656937, 77663192, 700614760, 6349125440, 57754842117, 527046644056, 4822774262296, 44235726874816, 406582639811581, 3743845040832376, 34529632747211560, 318931047174438720, 2949641596923575548, 27312107861301870368

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1+x)^(4*n-2)/(1-x).

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

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

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

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

D-finite with recurrence: 128*(4*n-7)*(2*n-3)*(4*n-9)*(22*n^2-17*n-15)*a(n-2) -8*(1892*n^5-11274*n^4+23326*n^3-18132*n^2+1323*n+2835)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^2-61*n+24)*a(n) = 0. - Georg Fischer, Aug 17 2025

a(n) ~ 2^(8*n - 9/2) / (sqrt(Pi*n) * 3^(3*n - 5/2)). - Vaclav Kotesovec, Oct 19 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-1,n-2*k). - Seiichi Manyama, Nov 11 2025

Mathematica

Table[Sum[Binomial[4*n-2, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 03 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n-2, k));
(Magma) [&+[Binomial(4*n-2, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
(Python)
from math import comb
def A387036(n): return 1 if n == 0 else sum(comb(4*n-2, k) for k in range(n+1)) # Aidan Chen, Jan 23 2026

Crossrefs

Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387037.

Cf. A002293, A262977.

Sequence in context: A362691 A138899 A147855 * A354327 A278333 A394239

Adjacent sequences: A387033 A387034 A387035 * A387037 A387038 A387039

Keyword

nonn

Author

Seiichi Manyama, Aug 13 2025

Status

approved