A346272 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).

1, 1, 3, 15, 115, 1196, 16282, 276158, 5713507, 140482000, 4047179258, 134447125418, 5097852537802, 218254682152053, 10469861372693621, 558373926672949031, 32908746221003292003, 2130712239317226923136, 150826951188229240683858, 11618459541824256750732794

Offset: 0

Author

Ilya Gutkovskiy, Jul 12 2021

Keywords

• nonn

Crossrefs

Cf. A023998, A061696, A124504, A346271.

Programs

Mathematica

nmax = 19; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^3/36], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = a[1] = 1; a[n_] := a[n] = n a[n - 1] + n (n - 1)^2 a[n - 2]/2 + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 19}]

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 + Sum_{n>=4} x^n / (n!)^2 ).

a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2 + (1/n) * Sum_{k=4..n} binomial(n,k)^2 * k * a(n-k).