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Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^(2*k-1) + A(x)) = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k + A(x))^2 / (1 + x^(2*k) + A(x))^2.
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%I #10 Jun 23 2024 22:06:57

%S 1,-4,18,-95,553,-3456,22657,-153716,1070043,-7599246,54840210,

%T -400989178,2964353647,-22119041245,166368440982,-1260046720460,

%U 9601545345559,-73557555227321,566224579674519,-4377328989747178,33970709342730943,-264555621945673287,2066853486071102281

%N Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^(2*k-1) + A(x)) = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k + A(x))^2 / (1 + x^(2*k) + A(x))^2.

%C Compare to identity: 1 + 2*Sum_{n>=0} x^(n^2) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n)^2 / (1 + x^(2*n))^2.

%H Paul D. Hanna, <a href="/A370344/b370344.txt">Table of n, a(n) for n = 1..250</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) B(x) = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k + A(x))^2 / (1 + x^(2*k) + A(x))^2.

%F (2) B(x) = Sum_{n>=0} Product_{k=1..n} (x^(2*k-1) + A(x)).

%F (3) B(x) = Sum_{n>=0} x^(n^2) / Product_{k=0..n} (1 - x^(2*k)*A(x)).

%F (4) B(x) = (x + A(x))/(1 + F(1)), where F(n) = -(x^(2*n+1) + A(x))/(1 + (x^(2*n+1) + A(x)) + F(n+1)), a continued fraction.

%e G.f.: A(x) = x - 4*x^2 + 18*x^3 - 95*x^4 + 553*x^5 - 3456*x^6 + 22657*x^7 - 153716*x^8 + 1070043*x^9 - 7599246*x^10 + 54840210*x^11 - 400989178*x^12 + ...

%e By definition, A = A(x) allows for the following expressions to equal

%e B(x) = 1 + (x + A) + (x + A)*(x^3 + A) + (x + A)*(x^3 + A)*(x^5 + A) + (x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A) + (x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A)*(x^9 + A) + ...

%e B(x) = (1 - x^2)*(1 + x + A)^2/(1 + x^2 + A)^2 * (1 - x^4)*(1 + x^2 + A)^2/(1 + x^4 + A)^2 * (1 - x^6)*(1 + x^3 + A)^2/(1 + x^6 + A)^2 * (1 - x^8)*(1 + x^4 + A)^2/(1 + x^8 + A)^2 * ...

%e where B(x) begins

%e B(x) = 1 + 2*x - 2*x^2 + 8*x^3 - 41*x^4 + 250*x^5 - 1584*x^6 + 10464*x^7 - 71330*x^8 + 498144*x^9 - 3546004*x^10 + 25635440*x^11 - 187708130*x^12 + ...

%o (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( sum(m=0,#A, prod(k=1,m, x^(2*k-1) + Ser(A)) ) - prod(m=1,#A, (1 - x^(2*m))*(1 + x^m + Ser(A))^2/(1 + x^(2*m) + Ser(A))^2 ), #A-1)); H=A; A[n+1]}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A370345, A370346, A370341.

%K sign

%O 1,2

%A _Paul D. Hanna_, Feb 16 2024