OFFSET
0,3
COMMENTS
Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*(n-1)*x + ((n+1)*x)^2).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..386
FORMULA
a(n) = Sum_{k=0..n} (-n)^k * binomial(n,k)^2.
a(n) = Sum_{k=0..n} (-n-1)^(n-k) * binomial(n,k) * binomial(n+k,k).
a(n) = Hypergeometric2F1(-n, -n, 1, -n). - Vaclav Kotesovec, May 07 2019
a(n) = n! * [x^n] exp((1 - n)*x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020
MAPLE
A307885:= n -> simplify(hypergeom([-n, -n], [1], -n));
seq(A307885(n), n = 0..30); # G. C. Greubel, May 31 2020
MATHEMATICA
Table[Hypergeometric2F1[-n, -n, 1, -n], {n, 0, 20}] (* Vaclav Kotesovec, May 07 2019 *)
PROG
(PARI) {a(n) = polcoef((1-(n-1)*x-n*x^2)^n, n)}
(PARI) {a(n) = sum(k=0, n, (-n)^k*binomial(n, k)^2)}
(PARI) {a(n) = sum(k=0, n, (-n-1)^(n-k)*binomial(n, k)*binomial(n+k, k))}
(Sage) [ hypergeometric([-n, -n], [1], -n).simplify_hypergeometric() for n in (0..30)] # G. C. Greubel, May 31 2020
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, May 02 2019
STATUS
approved