A218575 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^k)^n) ).

1, 1, 2, 5, 11, 26, 56, 125, 269, 578, 1228, 2600, 5447, 11366, 23575, 48664, 99950, 204383, 416196, 844299, 1706368, 3436555, 6898255, 13803732, 27539833, 54788703, 108703105, 215112006, 424628345, 836218453, 1643005834, 3221104945, 6301628342, 12303151494

Offset: 0

Author

Paul D. Hanna, Nov 02 2012

Keywords

• nonn

Comments

Compare to the dual g.f. of A219230:

exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*(1 + x^n)^k) ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 56*x^6 + 125*x^7 +...
where
log(A(x)) = x/(1*(1-x*(1+x))*(1-x^2*(1+x^2))*(1-x^3*(1+x^3))*...) +
x^2/(2*(1-x^2*(1+x)^2)*(1-x^4*(1+x^2)^2)*(1-x^6*(1+x^3)^2)*...) +
x^3/(3*(1-x^3*(1+x)^3)*(1-x^6*(1+x^2)^3)*(1-x^9*(1+x^3)^3)*...) +
x^4/(4*(1-x^4*(1+x)^4)*(1-x^8*(1+x^2)^4)*(1-x^12*(1+x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 120*x^6/6 + 274*x^7/7 + 527*x^8/8 + 1054*x^9/9 + 1973*x^10/10 + 3807*x^11/11 + 6824*x^12/12 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A218576, A218551, A219230.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,1/(1-x^(m*k)*(1+x^k)^m +x*O(x^n))))),n)}
for(n=0,40,print1(a(n),", "))