OFFSET
0,2
COMMENTS
Euler transform of A000984.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} A005430(d) ) * a(n-k).
a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (1/sqrt(1 - 4^(1-k)) - 1)/k = 0.0907540019413286886324751305813463657179452545... - Vaclav Kotesovec, May 10 2021
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
binomial(2*d, d), d=numtheory[divisors](j)), j=1..n)/n)
end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 13 2023
MATHEMATICA
nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d Binomial[2 d, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[Sum[(1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 09 2021
STATUS
approved