A321798 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4).

1, 1, 5, 23, 117, 636, 3607, 21106, 126489, 772468, 4789844, 30075937, 190851839, 1222000222, 7885041530, 51222338580, 334720178969, 2198755865424, 14511029102232, 96169424666028, 639757737711300, 4270520564506069, 28595671605541357, 192025292117465445, 1292866976587651519

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..199 from Ludovic Schwob)

Formula

a(n) = Sum_{k=0..n} (C(n,k)/(n-k+1)) * C(n+3*k-1,n-k).

a(n) ~ sqrt((1 - r*s)*(1 + 3*r*s)/(8*Pi*(8*s - 3))) / (n^(3/2) * r^(n+1)), where r = 0.139684805934917057093949761392656080860096066578... and s = 1.76437708701490464570032194388560298744432681226... are real roots of the system of equations s*(1 - r/(1 - r*s)^4) = 1, 4*r^2*s^2 = (1 - r*s)^5. - Vaclav Kotesovec, Nov 21 2018

Mathematica

a[n_] := Sum[Binomial[n, k] * Binomial[n + 3k - 1, n - k]/(n - k + 1), {k, 0,
n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2018 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+3*k-1, n-k)/(n-k+1)); \\ Michel Marcus, Nov 19 2018
(GAP) List([0..25], n->Sum([0..n], k->Binomial(n, k)/(n-k+1)*Binomial(n+3*k-1, n-k))); # Muniru A Asiru, Nov 24 2018

Crossrefs

Cf. A109081, A161797, A321799.

Sequence in context: A358607 A073596 A167248 * A005393 A193704 A294356

Adjacent sequences: A321795 A321796 A321797 * A321799 A321800 A321801

Keyword

nonn

Author

Ludovic Schwob, Nov 19 2018

Status

approved