A395782 Expansion of (Sum_{k>=0} binomial(4*k+1,k) * x^k)^4.

1, 20, 294, 3804, 45881, 529308, 5921820, 64781900, 696650580, 7391238736, 77568523822, 806770798500, 8328009400398, 85417717917288, 871280511426936, 8844707030389164, 89408619343690220, 900444396245910768, 9038422593653829688, 90455699063988431856, 902846885411835421209

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{i,j,k,l >= 0 and i+j+k+l=n} binomial(4*i+1,i) * binomial(4*j+1,j) * binomial(4*k+1,k) * binomial(4*l+1,l).

G.f.: B(x)^4 where B(x) is the g.f. of A052203.

a(n) = Sum_{k=0..n} 3^k * binomial(k+2,2) * binomial(4*n+7,n-k).

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

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (4*k+20) * 3^k * binomial(k+3,3) * binomial(4*n+7,n-1-k).

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+20) * 4^k * binomial(k+3,3) * binomial(4*n+3-k,n-1-k).

Prog

(PARI) a(n) = sum(k=0, n, 3^k*binomial(k+2, 2)*binomial(4*n+7, n-k));

Crossrefs

Cf. A052203, A308523, A395781.

Sequence in context: A190824 A016259 A271029 * A047795 A223956 A132168

Adjacent sequences: A395779 A395780 A395781 * A395783 A395784 A395785

Keyword

nonn,easy

Author

Seiichi Manyama, May 05 2026

Status

approved