A395780 Expansion of (Sum_{k>=0} binomial(3*k+1,k) * x^k)^4.

1, 16, 180, 1744, 15554, 131616, 1074072, 8536240, 66489687, 509765952, 3858791664, 28905439248, 214635119470, 1581949708352, 11585423108592, 84377281155168, 611551322732687, 4413481365001680, 31730521183547172, 227349945847349952, 1623986957273609304

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(3*i+1,i) * binomial(3*j+1,j) * binomial(3*k+1,k) * binomial(3*l+1,l).

G.f.: B(x)^4 where B(x) is the g.f. of A045721.

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

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

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+16) * 2^k * binomial(k+3,3) * binomial(3*n+7,n-1-k).

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (2*k+16) * 3^k * binomial(k+3,3) * binomial(3*n+3-k,n-1-k).

Prog

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

Crossrefs

Cf. A045721, A062236, A395779.

Cf. A045894.

Sequence in context: A227557 A269202 A269103 * A016305 A218895 A016909

Adjacent sequences: A395777 A395778 A395779 * A395781 A395782 A395783

Keyword

nonn,easy

Author

Seiichi Manyama, May 05 2026

Status

approved