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A369682 Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = theta_2(x^(1/2)) / x^(1/8).

%I A369682 #10 Feb 05 2024 04:22:20
%S A369682 1,4,12,38,112,332,972,2818,8098,23096,65418,184194,516080,1440334,
%T A369682 4008442,11135682,30912896,85835538,238601354,664447912,1854592214,
%U A369682 5189848462,14561237108,40954656118,115428662380,325847049200,920772219740,2602948470362,7356994944096,20779322594048
%N A369682 Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = theta_2(x^(1/2)) / x^(1/8).
%C A369682 Note: theta_2(x^(1/2)) / x^(1/8) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) - see A089799.
%H A369682 Paul D. Hanna, <a href="/A369682/b369682.txt">Table of n, a(n) for n = 0..325</a>
%H A369682 Vaclav Kotesovec, <a href="/A369682/a369682.jpg">Plot of a(n+1)/a(n) for n = 1..324</a>
%F A369682 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A369682 (1) Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^k + A(x)) = Sum_{n=-oo..+oo} x^(n*(n+1)/2).
%F A369682 (2) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) / Product_{k=1..n} (1 + x^k*A(x)) = x * Sum_{n=-oo..+oo} x^(n*(n+1)/2).
%F A369682 (3) theta_2(x^(1/2))/x^(1/8) = (1 + A(x))/(1 + F(1)), where F(n) = x*(x^n + A(x))/(1 - x*(x^n + A(x)) + F(n+1)), a continued fraction.
%F A369682 (4) x * theta_2(x^(1/2))/x^(1/8) = 1/(1 + F(1)), where F(n) = x^(n-1)/(1 - x^(n-1) + x^n*A + (1 + x^n*A) * F(n+1)), a continued fraction.
%e A369682 G.f.: A(x) = 1 + 4*x + 12*x^2 + 38*x^3 + 112*x^4 + 332*x^5 + 972*x^6 + 2818*x^7 + 8098*x^8 + 23096*x^9 + 65418*x^10 + 184194*x^11 + 516080*x^12 + ...
%e A369682 By definition, A = A(x) satisfies the sum of products
%e A369682 theta_2(x^(1/2))/x^(1/8) = (1 + A) - x*(1 + A)*(x + A) + x^2*(1 + A)*(x + A)*(x^2 + A) - x^3*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A) + x^4*(1 + A)*(x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) -+ ...
%e A369682 also, A = A(x) satisfies another sum of products
%e A369682 x*theta_2(x^(1/2))/x^(1/8) = 1 - 1/(1 + x*A) + x/((1 + x*A)*(1 + x^2*A)) - x^3/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)) + x^6/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)) - x^10/((1 + x*A)*(1 + x^2*A)*(1 + x^3*A)*(1 + x^4*A)*(1 + x^5*A)) +- ...
%e A369682 Further, A = A(x) satisfies the continued fraction given by
%e A369682 theta_2(x^(1/2))/x^(1/8) = (1 + A)/(1 + x*(x + A)/(1 - x*(x + A) + x*(x^2 + A)/(1 - x*(x^2 + A) + x*(x^3 + A)/(1 - x*(x^3 + A) + x*(x^4 + A)/(1 - x*(x^4 + A) + x*(x^5 + A)/(1 - x*(x^5 + A) + ...))))))
%e A369682 where theta_2(x^(1/2))/x^(1/8) = 2 + 2*x + 2*x^3 + 2*x^6 + 2*x^10 + 2*x^15 + 2*x^21 + ... + 2*x^(n*(n+1)/2) + ...
%o A369682 (PARI) {a(n) = my(A=[1], M = sqrtint(2*n)+1); for(i=1,n, A = concat(A,0);
%o A369682 A[#A] = polcoeff( sum(n=-M,M, x^(n*(n+1)/2) ) - sum(n=0,#A, (-1)^n * x^n * prod(k=0,n, x^k + Ser(A)) ), #A-1) ); H=A; A[n+1]}
%o A369682 for(n=0,30, print1(a(n),", "))
%Y A369682 Cf. A369683, A369684, A089799 (theta_2).
%K A369682 nonn
%O A369682 0,2
%A A369682 _Paul D. Hanna_, Feb 04 2024