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A370036 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).

%I A370036 #13 Feb 16 2025 08:34:06
%S A370036 1,6,35,199,1117,6335,37222,230809,1515784,10423684,73758799,
%T A370036 529151547,3815582934,27567473744,199625904531,1451286365478,
%U A370036 10610026385893,78068267016226,578088243024187,4304808678569939,32204405165738517,241805832191132439,1820963567348143772
%N A370036 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).
%C A370036 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370036 Paul D. Hanna, <a href="/A370036/b370036.txt">Table of n, a(n) for n = 1..326</a>
%H A370036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370036 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370036 (1) Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).
%F A370036 (2) Sum_{n=-oo..+oo} x^n * (x^n + 6*A(x))^(n-1) = 1 - 4*Sum_{n>=1} x^(n^2).
%F A370036 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 6*A(x))^n = 0.
%F A370036 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 6*x^n*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).
%F A370036 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 6*x^n*A(x))^(n+1) = 1 - 4*Sum_{n>=1} x^(n^2).
%F A370036 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 6*x^n*A(x))^n = 0.
%e A370036 G.f.: A(x) = x + 6*x^2 + 35*x^3 + 199*x^4 + 1117*x^5 + 6335*x^6 + 37222*x^7 + 230809*x^8 + 1515784*x^9 + 10423684*x^10 + 73758799*x^11 + 529151547*x^12 + ...
%e A370036 where
%e A370036 Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*x - 4*x^4 - 4*x^9 - 4*x^16 - 4*x^25 - 4*x^36 - 4*x^49 - ...
%e A370036 SPECIAL VALUES.
%e A370036 (V.1) Let A = A(exp(-Pi)) = 0.05816104948088020874729529058423242784366544822359858088...
%e A370036 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 6*A)^n = 3 - 2*Pi^(1/4)/gamma(3/4) = 0.82713037757338397...
%e A370036 (V.2) Let A = A(exp(-2*Pi)) = 0.001888597166059649200752082246148944967408910981759517793...
%e A370036 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 6*A)^n = 3 - 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9925302290245218...
%e A370036 (V.3) Let A = A(-exp(-Pi)) = -0.03427512499419794844050440831018295417511284891315471397...
%e A370036 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 6*A)^n = 3 - 2*(Pi/2)^(1/4)/gamma(3/4) = 1.172841723687766...
%e A370036 (V.4) Let A = A(-exp(-2*Pi)) = -0.001846744216948148769402996728724142172026226548695349349...
%e A370036 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 6*A)^n = 3 - 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.007469770878185...
%o A370036 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370036 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 6*Ser(A))^m ) - 1 + 4*sum(m=1,#A, x^(m^2) ), #A-1)/6 ); A[n+1]}
%o A370036 for(n=1,30, print1(a(n),", "))
%Y A370036 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370037, A370038, A370039, A370043.
%K A370036 nonn
%O A370036 1,2
%A A370036 _Paul D. Hanna_, Feb 10 2024