A395704 Expansion of g^4*(2*g-1)/(3-2*g)^4, where g = 1+x*g^3 is the g.f. of A001764.

1, 14, 144, 1308, 11105, 90360, 714084, 5524152, 42046371, 315964470, 2349970656, 17329573296, 126885222828, 923410872944, 6685074723420, 48177037031280, 345810084667287, 2473424154340794, 17635590347130816, 125387174972644140, 889218363195578241

Offset

0,2

Links

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

Formula

a(n) = (n+1) * A036829(n+1).

G.f.: (Sum_{k>=0} (4*k+1)/(3*k+1) * binomial(3*k+1,k) * x^k) * (Sum_{k>=0} binomial(3*k,k) * x^k)^3.

Sum_{k>=1} a(k-1) * x^k/k^2 = (1/3) * log( Sum_{k>=0} binomial(3*k,k) * x^k ).

a(n) = (n+1) * Sum_{k=0..n} binomial(3*k+1+l,k) * binomial(3*n-3*k-l,n-k) for every real number l.

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

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

Prog

(PARI) a(n) = (n+1)*sum(k=0, n, 2^(n-k)*binomial(3*n+2, k));

Crossrefs

Cf. A395705, A395706, A395707.

Cf. A001764, A036829, A378504.

Sequence in context: A238473 A016233 A276250 * A099914 A393329 A016278

Adjacent sequences: A395701 A395702 A395703 * A395705 A395706 A395707

Keyword

nonn

Author

Seiichi Manyama, May 04 2026

Status

approved