A335310 - OEIS

%I #12 Jun 01 2020 06:09:49

%S 1,1,-2,11,-74,477,-804,-84425,3315334,-102211207,3005297956,

%T -88338323709,2627003399164,-78764141488043,2341929797646648,

%U -66394419743289105,1609460569459689286,-18001777147777896975,-1625299659961386724524,196005371138608184827003

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

%H Seiichi Manyama, <a href="/A335310/b335310.txt">Table of n, a(n) for n = 0..386</a>

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

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

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

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

%t Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] (-n)^(n - k), {k, 0, n}], {n, 1, 19}]]

%t Table[SeriesCoefficient[1/Sqrt[1 + 2 (n - 2) x + n^2 x^2], {x, 0, n}], {n, 0, 19}]

%t Table[n! SeriesCoefficient[Exp[(2 - n) x] BesselI[0, 2 Sqrt[1 - n] x], {x, 0, n}], {n, 0, 19}]

%t Table[Hypergeometric2F1[-n, -n, 1, 1 - n], {n, 0, 19}]

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^2*(1-n)^k); \\ _Michel Marcus_, Jun 01 2020

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

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, May 31 2020