A346271 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).

1, 1, 2, 7, 41, 346, 3807, 53747, 952275, 20362552, 515112983, 15277888693, 523304644304, 20415373547609, 900219731675981, 44533809102813206, 2451041479421900803, 149140880201760643360, 9982798939295116151967, 731215136812226462200109, 58333374310397488522052976

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / (n!)^2 ).

a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} binomial(n,k)^2 * k * a(n-k).

Mathematica

nmax = 20; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 20}]

Crossrefs

Cf. A023998, A061696, A097514, A346272.

Sequence in context: A220896 A087804 A006677 * A101390 A113144 A320415

Adjacent sequences: A346268 A346269 A346270 * A346272 A346273 A346274

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 12 2021

Status

approved