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%I #5 Feb 24 2018 12:50:56
%S 1,1,2,5,16,58,228,949,4130,18633,86622,413106,2014489,10020342,
%T 50748198,261324021,1366804389,7255452421,39066835030,213287955245,
%U 1180397594359,6621150605830,37641036925947,216882646869800,1266660408267898,7499333882769716,45017192824063767,274030099624436499,1691811333997049888,10595032219552021063,67313254111562228356
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n+1))^n.
%C Compare g.f. to a g.f. C(x) of the Catalan sequence:
%C C(x) = Sum_{n>=0} x^n*(1 + x*C(x)^2)^n where C(x) = 1 + x*C(x)^2.
%H Paul D. Hanna, <a href="/A300042/b300042.txt">Table of n, a(n) for n = 0..100</a>
%F G.f. satisfies:
%F (1) A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n+1))^n.
%F (2) A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n*(n+1)) / (1 - x*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 58*x^5 + 228*x^6 + 949*x^7 + 4130*x^8 + 18633*x^9 + 86622*x^10 + 413106*x^11 + 2014489*x^12 + ...
%e such that
%e A(x) = 1 + x*(1+x*A(x)^2) + x^2*(1+x*A(x)^3)^2 + x^3*(1+x*A(x)^4)^3 + x^4*(1+x*A(x)^5)^4 + x^5*(1+x*A(x)^6)^5 + x^6*(1+x*A(x)^7)^6 + ...
%e The g.f. also satisfies the series identity:
%e A(x) = 1/(1-x) + x^2*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^6/(1-x*A(x)^2)^3 + x^6*A(x)^12/(1-x*A(x)^3)^4 + x^8*A(x)^20/(1-x*A(x)^4)^5 + x^10*A(x)^30/(1-x*A(x)^5)^6 + ...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(1+x*(A+x*O(x^n))^(m+1))^m)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k*(k+1))/(1 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A300041, A186998, A300043.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 24 2018
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