OFFSET
0,2
COMMENTS
In general, for k > 1, if e.g.f. satisfies A(x) = exp(x) + x*A(x)^k, then a(n) ~ sqrt(1 + LambertW((1 - 1/k)^k)) * (k-1)^(n - 1/2 + 1/(k-1)) * n^(n-1) / (k^(1/2 + 1/(k-1)) * exp(n) * LambertW((1 - 1/k)^k)^(n + 1/(k-1))).
FORMULA
a(n) ~ sqrt(1 + LambertW(81/256)) * 3^(n - 1/6) * n^(n-1) / (2^(5/3) * exp(n) * LambertW(81/256)^(n + 1/3)).
MATHEMATICA
Table[n! * Sum[(3*k+1)^(n-k-1) * Binomial[4*k, k] / (n-k)!, {k, 0, n}], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jun 01 2024
STATUS
approved