A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).

1, 1, 2, 4, 7, 14, 25, 44, 79, 137, 237, 408, 689, 1162, 1946, 3231, 5342, 8776, 14340, 23326, 37758, 60847, 97670, 156145, 248697, 394719, 624343, 984360, 1547187, 2424581, 3788730, 5904230, 9176723, 14226914, 22002523, 33947526, 52258177, 80268131, 123028407

Offset: 0

Author

Paul D. Hanna, Nov 02 2012

Keywords

• nonn

Comments

Compare to the dual g.f. of A219229:

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*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 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*x^10/10 +...
		

Links

• Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..2000 (terms 0..1000 from Paul D. Hanna)

Crossrefs

Cf. A219229, A218575, A218552, A218153.

Programs

PARI

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

Formula

Conjecture: a(n) ~ c * d^n, where d = A060006 = 1.3247179572447... is the real root of the equation d*(d^2-1) = 1 and c = 43328430766.390... . - Vaclav Kotesovec, Apr 09 2016