A303503 - OEIS

A303503

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

1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200

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OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..212

FORMULA

a(n) = A287318(n,n).

a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018

a(n) = A000984(n)*A033935(n). - Alois P. Heinz, Jan 30 2023

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))

end:

a:= n-> (2*n)!*b(n$2)/n!^2:

seq(a(n), n=0..17); # Alois P. Heinz, Jan 29 2023

MATHEMATICA

Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]

CROSSREFS

Main diagonal of A287318.

Cf. A000984, A002894, A002896, A033935, A039699, A287317, A361297.