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A331794
a(n) = Sum_{k=0..n} n^k * binomial(n+1,k) * binomial(n+1,k+1).
3
1, 4, 33, 400, 6285, 120456, 2714173, 70129984, 2040655401, 65956468600, 2342384363561, 90607200956064, 3789863084012629, 170370561866229648, 8188781210421259365, 418938023982360898816, 22724122083014879989905, 1302374806940392958470104
OFFSET
0,2
LINKS
FORMULA
a(n) = [x^n] 2/(1 - 2*(n+1)*x + ((n-1)*x)^2 + (1 - (n+1)*x) * sqrt(1 - 2*(n+1)*x + ((n-1)*x)^2)).
a(n) = (n+1) * 2F1(-1 - n, -n; 2; n), where 2F1 is the hypergeometric function. - Vaclav Kotesovec, Jan 26 2020
MATHEMATICA
Flatten[{1, Table[Sum[n^k * Binomial[n+1, k] * Binomial[n+1, k+1], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jan 26 2020 *)
Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 26 2020 *)
PROG
(PARI) {a(n) = sum(k=0, n, n^k*binomial(n+1, k)*binomial(n+1, k+1))}
(PARI) {a(n) = polcoef(2/(1-2*(n+1)*x+((n-1)*x)^2+(1-(n+1)*x)*sqrt(1-2*(n+1)*x+((n-1)*x)^2)), n)}
CROSSREFS
Sequence in context: A293193 A295256 A269926 * A156132 A215364 A370930
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 26 2020
STATUS
approved