A219262 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n / Product_{k>=1} (1 - x^(n*k)*A(x^k)^n) ).

1, 1, 3, 11, 43, 179, 772, 3446, 15737, 73315, 346869, 1662593, 8055433, 39390631, 194145569, 963490818, 4810435187, 24145458246, 121771909878, 616744946740, 3135657413451, 15997777629900, 81877262767137, 420263466778106, 2162874177974610, 11158378099471260, 57696623896231494

Offset: 0

Author

Paul D. Hanna, Nov 16 2012

Keywords

• nonn

Comments

Compare to the dual g.f. G(x) of A219263:

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

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 772*x^6 + 3446*x^7 +...
where
log(A(x)) = x*A(x)/1/(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
x^2*A(x)^2/2/((1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3/((1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4/((1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 121*x^4/4 + 601*x^5/5 + 2999*x^6/6 + 15247*x^7/7 + 78057*x^8/8 + 403279*x^9/9 + 2095425*x^10/10 +...
		

Crossrefs

Cf. A218551, A219260, A219263.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m/prod(k=1, n\m+1, 1-x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))