OFFSET
0,3
COMMENTS
Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..260
FORMULA
G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
(1) G_n(x) = Series_Reversion( x*(1 - x^n) ),
(2) G_n(x) = x + x*G_n(x)^(n+1),
(3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).
a(n) ~ c * 4^n / n^(3/2), where c = 0.19197348199... . - Vaclav Kotesovec, Nov 06 2014
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x*(1 - x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
Note that G_n(x) = x + x*G_n(x)^(n+1).
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, prod(k=1, m, serreverse(x*(1-x^k+x*O(x^n))))), n)}
for(n=0, 35, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 05 2012
STATUS
approved