A370026 - OEIS

%I #11 Feb 12 2024 19:32:52

%S 1,6,39,269,1917,13893,101830,753255,5614504,42110432,317474187,

%T 2403893757,18270065438,139305459960,1065183756535,8165168139498,

%U 62729216570805,482878316552298,3723769699813119,28762830132956421,222495155932381229,1723432870654770161,13366099075223254740

%N Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

%C A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

%H Paul D. Hanna, <a href="/A370026/b370026.txt">Table of n, a(n) for n = 1..401</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

%F (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^(n-1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

%F (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^n = 0.

%F (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

%F (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^(n+1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).

%F (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 6*A(x)*x^n)^(n+1) = 0.

%e G.f.: A(x) = x + 6*x^2 + 39*x^3 + 269*x^4 + 1917*x^5 + 13893*x^6 + 101830*x^7 + 753255*x^8 + 5614504*x^9 + 42110432*x^10 + 317474187*x^11 + ...

%e where

%e Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 - 8*x + 8*x^4 - 8*x^9 + 8*x^16 - 8*x^25 + 8*x^36 - 8*x^49 +- ...

%e SPECIAL VALUES.

%e (V.1) Let A = A(exp(-Pi)) = 0.05892551210473733684254468528377030200762221986684224912...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 6*A)^n = 4*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.65431655262446728562897...

%e (V.2) Let A = A(exp(-2*Pi)) = 0.001888624085511713374935799800784148455986111369097248489...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 6*A)^n = 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.985060458243628543159...

%e (V.3) Let A = A(-exp(-Pi)) = -0.03443859231795915470687740421610270983167641847531807729...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 6*A)^n = 4*Pi^(1/4)/gamma(3/4) - 3 = 1.3457392448532320583012...

%e (V.4) Let A = A(-exp(-2*Pi)) = -0.001846769433141026637620872576636896819075507182864480219...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 6*A)^n = 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3 = 1.01493954195095636419...

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

%o A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 6*Ser(A))^m ) - 1 - 8*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/6 ); A[n+1]}

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

%Y Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370027, A370028, A370029, A370042.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 09 2024