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A370027
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
12
1, 7, 52, 405, 3250, 26541, 219311, 1828657, 15360068, 129802889, 1102476535, 9403920685, 80507808128, 691425600548, 5954703569335, 51409228587355, 444806083780093, 3856115167020090, 33488422645226379, 291294693699275917, 2537471770952346625, 22133307405655321131
OFFSET
1,2
COMMENTS
A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
LINKS
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} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^(n-1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^(n+1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 7*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 7*x^2 + 52*x^3 + 405*x^4 + 3250*x^5 + 26541*x^6 + 219311*x^7 + 1828657*x^8 + 15360068*x^9 + 129802889*x^10 + 1102476535*x^11 + 9403920685*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 - 9*x + 9*x^4 - 9*x^9 + 9*x^16 - 9*x^25 + 9*x^36 - 9*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06265408791983395104830182276472061307372169283289177444...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 7*A)^n = (9*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.6111061217025256963...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892197774017068345453024031418945825808997896316975979...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 7*A)^n = (9*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.98319301552408211105...
(V.3) Let A = A(-exp(-Pi)) = -0.03328815108533045197898037729675109506494860109014140530...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 7*A)^n = (9*Pi^(1/4)/gamma(3/4) - 7)/2 = 1.388956650459886065588...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001843365127917378852723125074532830028319143070315792225...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 7*A)^n = (9*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 7)/2 = 1.0168069846948259097...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 7*Ser(A))^m ) - 1 - 9*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/7 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved