login
A337593
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 3^(k-1) * a(n-k).
5
1, 1, 5, 42, 573, 11226, 294804, 9946791, 417064365, 21187915362, 1278636342660, 90195692894451, 7338668846348844, 681052861293535251, 71405270562056271741, 8388541745045127600597, 1096298129481068449931085, 158383969954582566159384786, 25153555538082783169267336764
OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(3*x)) - 1) / 3).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 3^(n-1) * x^n / (n!)^2).
MATHEMATICA
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[3 x]] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!^2
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 02 2020
STATUS
approved