A219232 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).

1, 1, 2, 4, 11, 30, 92, 284, 918, 3005, 10043, 33943, 116138, 400862, 1395228, 4889389, 17240482, 61117789, 217709832, 778841527, 2797066886, 10080379573, 36444817306, 132147553180, 480444008087, 1751033068088, 6396352141777, 23414462628460, 85878613308907, 315556155264918

Offset: 0

Author

Paul D. Hanna, Nov 16 2012

Keywords

• nonn

Comments

Compare to the dual g.f. G(x) of A218552:

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

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 30*x^5 + 92*x^6 + 284*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
x^2/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
x^3/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 86*x^5/5 + 321*x^6/6 + 1128*x^7/7 + 4163*x^8/8 + 15172*x^9/9 + 56078*x^10/10 +...
		

Crossrefs

Cf. A218552, A219229, A219231, A219230, A218153, A218153.

Programs

PARI

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