OFFSET
1,8
FORMULA
G.f.: x^2 + x^3 / Product_{n>=1} (1 - x^n)^a(n).
a(1) = 0, a(2) = 1, a(3) = 1; a(n) = (1/(n - 3)) * Sum_{k=1..n-3} ( Sum_{d|k} d * a(d) ) * a(n-k).
a(n) ~ c * d^n / n^(3/2), where d = 1.646504994482771446591056040381099740295861136174688956979834656... and c = 0.8402317368556115946120005582458627329843217960728964299829... - Vaclav Kotesovec, Jul 06 2021
MAPLE
a:= proc(n) option remember; `if`(n<4, signum(n-1), add(a(n-k)*
add(d*a(d), d=numtheory[divisors](k)), k=1..n-3)/(n-3))
end:
seq(a(n), n=1..47); # Alois P. Heinz, Jul 01 2021
MATHEMATICA
nmax = 47; A[_] = 0; Do[A[x_] = x^2 + x^3 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 0; a[2] = 1; a[3] = 1; a[n_] := a[n] = (1/(n - 3)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 3}]; Table[a[n], {n, 1, 47}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jul 01 2021
STATUS
approved