A219230 - OEIS

%I #11 Nov 16 2012 00:44:00

%S 1,1,2,5,13,32,82,201,498,1214,2954,7117,17115,40880,97336,230699,

%T 545068,1283150,3011783,7047353,16445814,38275172,88859213,205796476,

%U 475539242,1096428621,2522704211,5792637135,13275381694,30367439045,69341077367,158059717986,359688534284

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

%C Compare to the dual g.f. of A218575:

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

%H Paul D. Hanna, <a href="/A219230/b219230.txt">Table of n, a(n) for n = 0..1000</a>

%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 32*x^5 + 82*x^6 + 201*x^7 +...

%e where

%e log(A(x)) = x/(1*(1-x*(1+x))*(1-x^2*(1+x)^2)*(1-x^3*(1+x)^3)*...) +

%e x^2/(2*(1-x^2*(1+x^2))*(1-x^4*(1+x^2)^2)*(1-x^6*(1+x^2)^3)*...) +

%e x^3/(3*(1-x^3*(1+x^3))*(1-x^6*(1+x^3)^2)*(1-x^9*(1+x^3)^3)*...) +

%e x^4/(4*(1-x^4*(1+x^4))*(1-x^8*(1+x^4)^2)*(1-x^12*(1+x^4)^3)*...) +...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 81*x^5/5 + 228*x^6/6 + 554*x^7/7 + 1399*x^8/8 + 3313*x^9/9 + 7843*x^10/10 +...

%o (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^m)^k +x*O(x^n))))), n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A218575, A219229, A219232.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 15 2012