A340971 - OEIS

%I #29 Nov 13 2021 05:36:42

%S 1,3,33,721,23649,1032801,56317969,3682424775,280767441537,

%T 24456613613401,2395993939827201,260764460901476643,

%U 31213273328323059169,4075382667781540713807,576394007453263029232497

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

%H Seiichi Manyama, <a href="/A340971/b340971.txt">Table of n, a(n) for n = 0..322</a>

%F a(n) = [x^n] 1/sqrt((1-x)*(1-(4*n+1)*x)).

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

%F a(n) = n! * [x^n] exp((2*n+1)*x) * BesselI(0,2*n*x). - _Ilya Gutkovskiy_, Feb 01 2021

%F a(n) ~ exp(1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Nov 13 2021

%t a[0] = 1; a[n_] := Sum[n^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 15, 0] (* _Amiram Eldar_, Feb 01 2021 *)

%o (PARI) {a(n) = sum(k=0, n, n^k*binomial(n, k)*binomial(2*k, k))}

%o (PARI) {a(n) = polcoef(1/sqrt((1-x)*(1-(4*n+1)*x)+x*O(x^n)), n)}

%o (PARI) {a(n) = polcoef((1+(2*n+1)*x+(n*x)^2)^n, n)}

%Y Main diagonal of A340970.

%Y Cf. A338979, A340972.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 01 2021