A352658 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).

1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359

Offset: 0

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

• nonn

Crossrefs

Cf. A000217, A002411, A023998, A279361, A336227, A337591.

Programs

Mathematica

a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 Binomial[k + 1, 2] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(x BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} binomial(n+1,2) * x^n / n!^2 ).