OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{n>=0, k=0..n*(n+1)/2} A053632(n,k)*x^n*(1+x)^k, where A053632(n,k) = number of partitions of k into distinct parts <= n.
G.f.: 1/(G(0) - 2*x) where G(k) = 1 + x + x*(1 + x)^k - x*(1 + (1 + x)^(k+1))/G(k+1); (recursively defined continued fraction; G(0)=2*x). - Sergei N. Gladkovskii, Dec 15 2012
G.f.: Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / ( Product_{k=0..n} 1 - x*(1+x)^k ). - Paul D. Hanna, Nov 09 2020
EXAMPLE
G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 44*x^4 + 151*x^5 + 560*x^6 +...
such that, by definition,
A(x) = 1 + x*(1 + (1+x)) + x^2*(1 + (1+x))*(1 + (1+x)^2) + x^3*(1 + (1+x))*(1 + (1+x)^2)*(1 + (1+x)^3) +...
PROG
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (1+(1+x)^k)+x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 19 2012
STATUS
approved