OFFSET
0,3
COMMENTS
Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1+4*n)*x^2).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (-n)^k * binomial(n,2*k) * binomial(2*k,k).
a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(-n)*x). - Ilya Gutkovskiy, May 31 2020
a(n) = Hypergeometric2F1(-n/2, (1-n)/2; 1; -4*n). - G. C. Greubel, May 31 2020
MAPLE
A307862:= n -> simplify(hypergeom([-n/2, (1-n)/2], [1], -4*n));
seq(A307862(n), n = 0..30); # G. C. Greubel, May 31 2020
MATHEMATICA
a[n_]:= SeriesCoefficient[(1 +x -n*x^2)^n, {x, 0, n}]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, May 31 2020 *)
PROG
(PARI) {a(n) = polcoef((1+x-n*x^2)^n, n)}
(PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, k)*binomial(n-k, k))}
(PARI) {a(n) = sum(k=0, n\2, (-n)^k*binomial(n, 2*k)*binomial(2*k, k))}
(Sage) [ hypergeometric([-n/2, (1-n)/2], [1], -4*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