A395705 Expansion of g^5*(3*g-2)/(4-3*g)^4, where g = 1+x*g^4 is the g.f. of A002293.

1, 20, 291, 3728, 44580, 510540, 5675922, 61752384, 660882060, 6981874000, 72993120299, 756574948464, 7785505959244, 79626147107976, 810083441973780, 8203662416077056, 82743485273450172, 831590904057001200, 8331140468822130556, 83226329338216485440, 829278305196569713296

Offset

0,2

Links

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

Formula

a(n) = (n+1) * A308523(n+1).

G.f.: (Sum_{k>=0} (3*k+1)/(2*k+1) * binomial(4*k+2,k) * x^k) * (Sum_{k>=0} binomial(4*k,k) * x^k)^3.

Sum_{k>=1} a(k-1) * x^k/k^2 = (1/4) * log( Sum_{k>=0} binomial(4*k,k) * x^k ).

a(n) = (n+1) * Sum_{k=0..n} binomial(4*k+2+l,k) * binomial(4*n-4*k-l,n-k) for every real number l.

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

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

Prog

(PARI) a(n) = (n+1)*sum(k=0, n, 3^(n-k)*binomial(4*n+3, k));

Crossrefs

Cf. A395704, A395706, A395707.

Cf. A002293, A308523, A395664, A395709.

Sequence in context: A329710 A125477 A384326 * A016261 A016309 A190824

Adjacent sequences: A395702 A395703 A395704 * A395706 A395707 A395708

Keyword

nonn

Author

Seiichi Manyama, May 04 2026

Status

approved