A294511 - OEIS

%I #13 Jan 06 2023 15:58:47

%S 1,1,6,57,752,12730,263172,6427239,181073552,5780683422,206236365320,

%T 8131806499806,351152348814720,16481622338583844,835440526931000200,

%U 45483669552645222975,2646983711809388956320,163980758725017626674422,10774158375350926668928248

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

%H Alois P. Heinz, <a href="/A294511/b294511.txt">Table of n, a(n) for n = 0..369</a>

%F a(n) = A294498(n,n).

%F a(n) ~ c * d^n * n^n, where d = 1.38107962281156411118368063513315508389931... and c = 0.822562099033048851679532590998951521... - _Vaclav Kotesovec_, Nov 13 2017

%F a(n) mod 2 = 1 <=> n in { A000225 }. - _Alois P. Heinz_, Jan 06 2023

%p C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:

%p b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, C(n),

%p       (h-> add(binomial(n, j)*b(j, h)*b(n-j, k-h), j=0..n))(iquo(k, 2))))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 06 2023

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

%Y Main diagonal of A294498.

%Y Cf. A000108, A000225.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 01 2017