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%I #3 Jul 26 2012 12:59:35
%S 1,1,8,116,1972,36682,722098,14784834,311629580,6716892893,
%T 147372681787,3280609461927,73912217824094,1682234535898788,
%U 38621258859241912,893358073179541313,20800314016777824187,487100732909778007223,11465386711990265812207
%N G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(10*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.
%F G.f. satisfies: 1+x = A(y) where y is a 25-degree polynomial in x and is the g.f. of row 5 in triangle A214690.
%F G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+10)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).
%e G.f.: A(x) = 1 + x + 8*x^2 + 116*x^3 + 1972*x^4 + 36682*x^5 + 722098*x^6 +...
%e The g.f. satisfies:
%e x = (A(x)-1)/A(x)^11 + (A(x)-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^39 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^56 +
%e (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^75 +...
%o (PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 8*x^2 + 12*x^3 + 108*x^4 - 218*x^5 - 1938*x^6 - 834*x^7 + 27124*x^8 + 136919*x^9 + 393601*x^10 +
%o 809873*x^11 + 1288950*x^12 + 1646268*x^13 + 1720788*x^14 + 1487263*x^15 + 1067345*x^16 + 635682*x^17 + 312646*x^18 + 125761*x^19 + 40734*x^20 +
%o 10373*x^21 + 2001*x^22 + 275*x^23 + 24*x^24 + x^25 +x^2*O(x^n)), n))}
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(10*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. A214690, A214692, A214693, A214694, A209441 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 26 2012
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