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A182970 G.f.: A(x) = Product_{n>=1} 1/(1 - A_n(x)^n) where A_n(x) denotes the n-th iteration of A(x): A_n(x) = A_{n-1}(A(x)) with A_0(x)=x.

%I A182970 #13 Feb 11 2025 00:56:32
%S A182970 1,1,3,12,61,365,2477,18566,150940,1314016,12135518,118077620,
%T A182970 1204031386,12814054072,141872524160,1629774749836,19383459694769,
%U A182970 238243063976805,3021510752477432,39488027180606978,531178015089101579,7346877516617129889
%N A182970 G.f.: A(x) = Product_{n>=1} 1/(1 - A_n(x)^n) where A_n(x) denotes the n-th iteration of A(x): A_n(x) = A_{n-1}(A(x)) with A_0(x)=x.
%F A182970 G.f.: A(x) = x*exp( Sum_{n>=1} Sum_{d|n} d*A_d(x)^n/n ) where A_n(x) denotes the n-th iteration of A(x).
%e A182970 G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 61*x^5 + 365*x^6 +...
%e A182970 Let A_n(x) denote the n-th iteration of g.f. A(x), then
%e A182970 the logarithm of A(x)/x begins:
%e A182970 log(A(x)/x) = A(x) + [A(x)^2 + 2*A_2(x)^2]/2 + [A(x)^3 + 3*A_3(x)^3]/3 + [A(x)^4 + 2*A_2(x)^4 + 4*A_4(x)^4]/4 + [A(x)^5 + 5*A_5(x)^5]/5 +...
%e A182970 Explicitly,
%e A182970 log(A(x)/x) = x + 5*x^2/2 + 28*x^3/3 + 189*x^4/4 + 1431*x^5/5 + 11858*x^6/6 + 105533*x^7/7 + 996541*x^8/8 + 9901306*x^9/9 + 102895485*x^10/10 +...
%e A182970 The initial iterations of A(x) begin:
%e A182970   A_2(x) = A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 236*x^5 + 1571*x^6 +...
%e A182970   A_3(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 613*x^5 + 4586*x^6 +...
%e A182970   A_4(x) = x + 4*x^2 + 24*x^3 + 168*x^4 + 1304*x^5 + 10926*x^6 +...
%e A182970   A_5(x) = x + 5*x^2 + 35*x^3 + 280*x^4 + 2445*x^5 + 22775*x^6 +...
%e A182970   A_6(x) = x + 6*x^2 + 48*x^3 + 432*x^4 + 4196*x^5 + 43105*x^6 +...
%e A182970   A_7(x) = x + 7*x^2 + 63*x^3 + 630*x^4 + 6741*x^5 + 75796*x^6 +...
%e A182970   A_8(x) = x + 8*x^2 + 80*x^3 + 880*x^4 + 10288*x^5 + 125756*x^6 +...
%e A182970 The g.f. equals the product:
%e A182970 A(x) = x / Product_{n>=1} (1 - A_n(x)^n) = x/((1 - A(x))*(1 - A(A(x))^2)*(1 - A(A(A(x)))^3)*(1 - A(A(A(A(x))))^4)*...)
%e A182970 where A_n(x) equals the n-th iteration of A(x).
%o A182970 (PARI) /* n-th Iteration of a function: */
%o A182970 {ITERATE(n,F,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
%o A182970 /* G.f.: */
%o A182970 {a(n)=local(F=x+x^2+x*O(x^n));for(i=0,n,F=x*exp(sum(m=1,n+1,1/m*sumdiv(m,d,d*ITERATE(d,F,n)^m))));polcoeff(F,n)}
%o A182970 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(k=1,n,1-ITERATE(k,A,n)^k));polcoeff(A,n)}
%K A182970 nonn
%O A182970 1,3
%A A182970 _Paul D. Hanna_, Dec 18 2010
%E A182970 Name changed by _Paul D. Hanna_, Dec 19 2010