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%I #11 Nov 16 2012 19:33:57
%S 1,1,2,4,11,30,92,284,918,3005,10043,33943,116138,400862,1395228,
%T 4889389,17240482,61117789,217709832,778841527,2797066886,10080379573,
%U 36444817306,132147553180,480444008087,1751033068088,6396352141777,23414462628460,85878613308907,315556155264918
%N G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).
%C Compare to the dual g.f. G(x) of A218552:
%C G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^k)^n) ).
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 30*x^5 + 92*x^6 + 284*x^7 +...
%e where
%e log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
%e x^2/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
%e x^3/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
%e x^4/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
%e Explicitly,
%e 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 +...
%o (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)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A218552, A219229, A219231, A219230, A218153, A218153.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 16 2012
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