OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..386
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[(-1)^k * 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
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jan 26 2020
STATUS
approved