This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369684 #13 Feb 05 2024 04:29:31
%S A369684 1,-4,6,-14,27,-54,110,-217,445,-905,1863,-3858,7986,-16599,34438,
%T A369684 -71445,148075,-306551,634469,-1312707,2716636,-5624353,11649994,
%U A369684 -24144393,50059148,-103820127,215351391,-446713118,926604822,-1921881919,3985904949,-8266207127,17142752984
%N A369684 Expansion of g.f. A(x) satisfying Sum_{n>=0} x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = theta_4(x).
%C A369684 Note: theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2) - see A002448.
%H A369684 Paul D. Hanna, <a href="/A369684/b369684.txt">Table of n, a(n) for n = 0..325</a>
%F A369684 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A369684 (1) Sum_{n>=0} x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369684 (2) Sum_{n>=0} x^(n*(n+1)) / Product_{k=0..n} (1 - x^(2*k+1)*A(x)) = 1 + x * Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369684 (3) theta_4(x) = (x + A(x))/(1 + F(1)), where F(n) = -x*(x^(2*n+1) + A(x))/(1 + x*(x^(2*n+1) + A(x)) + F(n+1)), a continued fraction.
%F A369684 (4) 1 + x*theta_4(x) = 1/((1 - x*A(x))*(1 + F(1))), where F(n) = -x^(2*n) / (1 + x^(2*n) - x^(2*n+1)*A + (1 - x^(2*n+1)*A)*F(n+1)), a continued fraction.
%e A369684 G.f.: A(x) = 1 - 4*x + 6*x^2 - 14*x^3 + 27*x^4 - 54*x^5 + 110*x^6 - 217*x^7 + 445*x^8 - 905*x^9 + 1863*x^10 - 3858*x^11 + 7986*x^12 + ...
%e A369684 By definition, A = A(x) satisfies the sum of products
%e A369684 theta_4(x) = (x + A) + x*(x + A)*(x^3 + A) + x^2*(x + A)*(x^3 + A)*(x^5 + A) + x^3*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A) + x^4*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A)*(x^9 + A) + ...
%e A369684 also, A = A(x) satisfies another sum of products
%e A369684 1 + x*theta_4(x) = 1/(1 - x*A) + x^2/((1 - x*A)*(1 - x^3*A)) + x^6/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)) + x^12/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)*(1 - x^7*A)) + x^20/((1 - x*A)*(1 - x^3*A)*(1 - x^5*A)*(1 - x^7*A)*(1 - x^9*A)) + ...
%e A369684 Further, A = A(x) satisfies the continued fraction given by
%e A369684 theta_4(x) = (x + A)/(1 - x*(x^3 + A)/(1 + x*(x^3 + A) - x*(x^5 + A)/(1 + x*(x^5 + A) - x*(x^7 + A)/(1 + x*(x^7 + A) - x*(x^9 + A)/(1 + x*(x^9 + A) - x*(x^11 + A)/(1 + ...))))))
%e A369684 where theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 -+ ... + (-1)^n*2*x^(n^2) + ...
%e A369684 SPECIAL VALUES.
%e A369684 (V.1) A(exp(-Pi)) = 0.83730587071032100304130860721328123647753330074821532779...
%e A369684 where Sum_{n>=0} exp(-n*Pi) * Product_{k=0..n} (exp(-(2*k+1)*Pi) + A(exp(-Pi))) = (Pi/2)^(1/4)/gamma(3/4) = 0.91357913815611682140...
%e A369684 (V.2) A(exp(-2*Pi)) = 0.992551062280675678319013190897648447080249317782864483...
%e A369684 where Sum_{n>=0} exp(-2*n*Pi) * Product_{k=0..n} (exp(-2*(2*k+1)*Pi) + A(exp(-2*Pi))) = (Pi/2)^(1/4)/gamma(3/4) * 2^(1/8) = 0.99626511456090713578995...
%e A369684 (V.3) A(exp(-4*Pi)) = 0.99998605070354391051731649267915065106164831758249400...
%e A369684 where Sum_{n=-oo..+oo} exp(-4*n*Pi) * Product_{k=0..n} (exp(-4*(2*k+1)*Pi) + A(exp(-4*Pi))) = Pi^(1/4)/gamma(3/4) * (sqrt(2) + 1)^(1/4)/2^(7/16) = 0.99999302531528758200931...
%e A369684 (V.4) A(exp(-10*Pi)) = 0.9999999999999091559572670393411928665159764707765156...
%e A369684 where Sum_{n=-oo..+oo} exp(-10*n*Pi) * Product_{k=0..n} (exp(-10*(2*k+1)*Pi) + A(exp(-10*Pi))) = Pi^(1/4)/gamma(3/4) * 2^(7/8)/((5^(1/4) - 1)*sqrt(5*sqrt(5) + 5)) = 0.99999999999995457797863...
%o A369684 (PARI) {a(n) = my(A=[1], M = sqrtint(2*n)+1); for(i=1,n, A = concat(A,0);
%o A369684 A[#A] = polcoeff( sum(n=-M,M, (-1)^n * x^(n^2) ) - sum(n=0,#A, x^n * prod(k=0,n, x^(2*k+1) + Ser(A)) ), #A-1) ); H=A; A[n+1]}
%o A369684 for(n=0,40, print1(a(n),", "))
%Y A369684 Cf. A369683, A369682, A369671, A002448 (theta_4).
%K A369684 sign
%O A369684 0,2
%A A369684 _Paul D. Hanna_, Feb 04 2024