A294511 a(n) = n! * [x^n] exp(2*n*x)*(BesselI(0,2*x) - BesselI(1,2*x))^n.

1, 1, 6, 57, 752, 12730, 263172, 6427239, 181073552, 5780683422, 206236365320, 8131806499806, 351152348814720, 16481622338583844, 835440526931000200, 45483669552645222975, 2646983711809388956320, 163980758725017626674422, 10774158375350926668928248

Offset: 0

Author

Ilya Gutkovskiy, Nov 01 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..369

Crossrefs

Main diagonal of A294498.

Cf. A000108, A000225.

Programs

Maple

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, C(n),
      (h-> add(binomial(n, j)*b(j, h)*b(n-j, k-h), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 06 2023

Mathematica

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

Formula

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

a(n) ~ c * d^n * n^n, where d = 1.38107962281156411118368063513315508389931... and c = 0.822562099033048851679532590998951521... - Vaclav Kotesovec, Nov 13 2017

a(n) mod 2 = 1 <=> n in { A000225 }. - Alois P. Heinz, Jan 06 2023