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).

1, 6, 35, 199, 1117, 6335, 37222, 230809, 1515784, 10423684, 73758799, 529151547, 3815582934, 27567473744, 199625904531, 1451286365478, 10610026385893, 78068267016226, 578088243024187, 4304808678569939, 32204405165738517, 241805832191132439, 1820963567348143772

Offset

1,2

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..326

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

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

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

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

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

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

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

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

Example

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 + ...
where
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 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05816104948088020874729529058423242784366544822359858088...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 6*A)^n = 3 - 2*Pi^(1/4)/gamma(3/4) = 0.82713037757338397...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888597166059649200752082246148944967408910981759517793...
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...
(V.3) Let A = A(-exp(-Pi)) = -0.03427512499419794844050440831018295417511284891315471397...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 6*A)^n = 3 - 2*(Pi/2)^(1/4)/gamma(3/4) = 1.172841723687766...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846744216948148769402996728724142172026226548695349349...
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...

Prog

(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
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]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370037, A370038, A370039, A370043.

Sequence in context: A230713 A131435 A209179 * A081105 A079027 A289784

Adjacent sequences: A370033 A370034 A370035 * A370037 A370038 A370039

Keyword

nonn

Author

Paul D. Hanna, Feb 10 2024

Status

approved