OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)
FORMULA
a(n) = Sum_{i+j=n} [x^i*y^j] 1/2 * Product_{k,m>=0} (1+x^k*y^m).
G.f.: Product_{k>=1} (1+x^k)^(k+1). - Vaclav Kotesovec, Mar 07 2015
a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (1296*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * 3^(4/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(2 - x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Aug 11 2018
EXAMPLE
a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..42); # Alois P. Heinz, Sep 19 2019
MATHEMATICA
nmax=50; CoefficientList[Series[Product[(1+x^k)^(k+1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2015 *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 22 2012
STATUS
approved
