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%I #14 Dec 01 2014 03:21:31
%S 1,1,1,4,19,107,671,4600,34218,276415,2439426,23724674,256361107,
%T 3091554768,41560590331,618957882104,10119509431084,179887355572358,
%U 3446915545155744,70686674091569072,1542478858735415921,35650141769790146478,869385516566240903091,22299067147713040916568
%N G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(3*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).
%C Compare the g.f. to the identity:
%C G(x) = Sum_{n>=0} 1/G(x)^(2*n) * Product_{k=1..n} (1 - 1/G(x)^(2*k-1))
%C which holds for all power series G(x) such that G(0)=1.
%H Paul D. Hanna and Vaclav Kotesovec, <a href="/A249934/b249934.txt">Table of n, a(n) for n = 0..240</a> (first 100 terms from Paul D. Hanna)
%F G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+3)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
%F a(n) ~ exp(Pi^2/24) * 12^n * n^(n-1) / (sqrt(6) * exp(n) * Pi^(2*n-1)). - _Vaclav Kotesovec_, Dec 01 2014
%e A(x) = 1 + x + x^2 + 4*x^3 + 19*x^4 + 107*x^5 + 671*x^6 + 4600*x^7 + 34218*x^8 +...
%e The g.f. satisfies:
%e x = (A(x)-1)/A(x)^4 + (A(x)-1)*(A(x)^3-1)/A(x)^10 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^18 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^28 +
%e (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^40 +...
%t nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF^3,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* More efficient than PARI program, _Vaclav Kotesovec_, Nov 30 2014 *)
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
%o A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(3*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A214692.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 27 2014
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