A370029 - OEIS

%I #10 Feb 12 2024 19:34:28

%S 1,9,84,809,7974,79863,809131,8270199,85126516,881290445,9167900511,

%T 95763822969,1003839653480,10554997636854,111280621221379,

%U 1176017223671139,12454545436154097,132149953604522106,1404591515239624671,14952277258870348035,159396459604398283553

%N Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 9*A(x))^n = 1 + 11*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="/A370029/b370029.txt">Table of n, a(n) for n = 1..262</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 + 9*A(x))^n = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).

%F (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 9*A(x))^(n-1) = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).

%F (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 9*A(x))^n = 0.

%F (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 9*A(x)*x^n)^n = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).

%F (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 9*A(x)*x^n)^(n+1) = 1 + 11*Sum_{n>=1} (-1)^n * x^(n^2).

%F (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 9*A(x)*x^n)^(n+1) = 0.

%e G.f.: A(x) = x + 9*x^2 + 84*x^3 + 809*x^4 + 7974*x^5 + 79863*x^6 + 809131*x^7 + 8270199*x^8 + 85126516*x^9 + 881290445*x^10 + 9167900511*x^11 + 95763822969*x^12 + ...

%e where

%e Sum_{n=-oo..+oo} (-1)^n * (x^n + 9*A(x))^n = 1 - 11*x + 11*x^4 - 11*x^9 + 11*x^16 - 11*x^25 + 11*x^36 - 11*x^49 +- ...

%e SPECIAL VALUES.

%e (V.1) Let A = A(exp(-Pi)) = 0.07175219834164736620386280600888962717215573957821859403...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 9*A)^n = (11*(Pi/2)^(1/4)/gamma(3/4) - 9)/2 = 0.5246852598586425177...

%e (V.2) Let A = A(exp(-2*Pi)) = 0.001899385878782719362352219788087550672661478114904760835...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 9*A)^n = (11*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 9)/2 = 0.97945813008498924684...

%e (V.3) Let A = A(-exp(-Pi)) = -0.03120408533767785789845054540220571847531668789278074466...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 9*A)^n = (11*Pi^(1/4)/gamma(3/4) - 9)/2 = 1.475391461673194080164...

%e (V.4) Let A = A(-exp(-2*Pi)) = -0.001836594032195533189068390983153367342311468510211476381...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 9*A)^n = (11*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 9)/2 = 1.0205418701825650...

%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 + 9*Ser(A))^m ) - 1 - 11*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/9 ); A[n+1]}

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

%Y Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370042.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 09 2024