A386919 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k,n-k).

1, 6, 58, 624, 7050, 81926, 969640, 11624976, 140708682, 1715727090, 21043480458, 259331888712, 3208566672792, 39830312782344, 495853462219600, 6188170518911264, 77393543796042570, 969771226630919754, 12172039459124750062, 153006230384961477600, 1925930502301667496250

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Formula

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

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

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

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

a(n) ~ sqrt(1 + 3/sqrt(17)) * ((51*sqrt(17) - 107)/8)^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 19 2025

Mathematica

Table[Sum[Binomial[4*n, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 10 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k, n-k));
(Magma) [&+[Binomial(4*n, k) * Binomial(2*n-k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 10 2025

Crossrefs

Cf. A066381, A386918, A386920.

Cf. A385639.

Sequence in context: A292574 A354228 A073848 * A388535 A141382 A212426

Adjacent sequences: A386916 A386917 A386918 * A386920 A386921 A386922

Keyword

nonn

Author

Seiichi Manyama, Aug 08 2025

Status

approved