A335310 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * (-n)^(n-k).

1, 1, -2, 11, -74, 477, -804, -84425, 3315334, -102211207, 3005297956, -88338323709, 2627003399164, -78764141488043, 2341929797646648, -66394419743289105, 1609460569459689286, -18001777147777896975, -1625299659961386724524, 196005371138608184827003

Offset: 0

Author

Ilya Gutkovskiy, May 31 2020

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A098332, A116091, A126869, A307884, A307885, A331657, A335309.

Programs

Mathematica

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] (-n)^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[SeriesCoefficient[1/Sqrt[1 + 2 (n - 2) x + n^2 x^2], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[(2 - n) x] BesselI[0, 2 Sqrt[1 - n] x], {x, 0, n}], {n, 0, 19}]
Table[Hypergeometric2F1[-n, -n, 1, 1 - n], {n, 0, 19}]

PARI

a(n) = sum(k=0, n, binomial(n,k)^2*(1-n)^k); \\ Michel Marcus, Jun 01 2020

Formula

a(n) = central coefficient of (1 - (n - 2)*x - (n - 1)*x^2)^n.

a(n) = [x^n] 1 / sqrt(1 + 2*(n - 2)*x + n^2*x^2).

a(n) = n! * [x^n] exp((2 - n)*x) * BesselI(0,2*sqrt(1 - n)*x).

a(n) = Sum_{k=0..n} binomial(n,k)^2 * (1-n)^k.