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
KEYWORD
nonn
AUTHOR
Ludovic Schwob, Nov 19 2018
STATUS
approved