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%I #9 May 21 2019 23:47:13
%S 1,2,5,15,59,262,1307,7074,41012,252187,1632799,11074271,78360644,
%T 576612899,4400858604,34762324434,283657506679,2387255705823,
%U 20692499748472,184498910151279,1690257693844243,15894461099811120,153272602343966985,1514370059327255381,15317844239550849137,158501683635111855424,1676615643571796233437,18117887771586127697132,199886514026342226648647
%N G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} x^n * ((1+x)^n - A(x))^n, where A(0) = 0.
%H Paul D. Hanna, <a href="/A325575/b325575.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F (1) 1/(1-x) = Sum_{n>=0} x^n * ((1+x)^n - A(x))^n.
%F (2) 1/(1-x) = Sum_{n>=0} x^n * (1+x)^(n^2) / (1 + x*(1+x)^n*A(x))^(n+1).
%e G.f.: A(x) = x + 2*x^2 + 5*x^3 + 15*x^4 + 59*x^5 + 262*x^6 + 1307*x^7 + 7074*x^8 + 41012*x^9 + 252187*x^10 + 1632799*x^11 + 11074271*x^12 + ...
%e such that
%e 1/(1-x) = 1 + x*((1+x) - A(x)) + x^2*((1+x)^2 - A(x))^2 + x^3*((1+x)^3 - A(x))^3 + x^4*((1+x)^4 - A(x))^4 + x^5*((1+x)^5 - A(x))^5 + x^6*((1+x)^6 - A(x))^6 + ...
%e also
%e 1/(1-x) = 1/(1 + x*A(x)) + x*(1+x)/(1 + x*(1+x)*A(x))^2 + x^2*(1+x)^4/(1 + x*(1+x)^2*A(x))^3 + x^3*(1+x)^9/(1 + x*(1+x)^3*A(x))^4 + x^4*(1+x)^16/(1 + x*(1+x)^4*A(x))^5 + x^5*(1+x)^25/(1 + x*(1+x)^5*A(x))^6 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff(sum(m=0,#A, x^m*((1+x)^m - x*Ser(A))^m ),#A+1));A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A325577, A307940.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 16 2019
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