A219229 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

1, 1, 2, 4, 9, 20, 45, 97, 217, 476, 1043, 2256, 4890, 10540, 22695, 48614, 103856, 221103, 469738, 995299, 2104514, 4439410, 9345471, 19631897, 41162937, 86148416, 179987656, 375411910, 781773332, 1625473202, 3374684346, 6996168098, 14483902403, 29945326755

Offset

0,3

Comments

Compare to the dual g.f. of A218576:

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

Links

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 45*x^6 + 97*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^2)^3)*...) +
x^3/3*((1+x^3*(1+x^3))*(1+x^6*(1+x^3)^2)*(1+x^9*(1+x^3)^3)*...) +
x^4/4*((1+x^4*(1+x^4))*(1+x^8*(1+x^4)^2)*(1+x^12*(1+x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 46*x^5/5 + 111*x^6/6 + 232*x^7/7 + 555*x^8/8 + 1204*x^9/9 + 2608*x^10/10 +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*prod(k=1, n\m, (1+x^(m*k)*(1+x^m+x*O(x^n))^k )))), n)}
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Cf. A218576, A219230, A219231.

Sequence in context: A214952 A360871 A199296 * A091620 A208738 A144686

Adjacent sequences: A219226 A219227 A219228 * A219230 A219231 A219232

Keyword

nonn

Author

Paul D. Hanna, Nov 15 2012

Status

approved