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A370038
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
13
1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, 118834966, 1014912939, 8876489811, 79106007766, 714758437500, 6521121292423, 59905861779190, 553172777516749, 5129986605394544, 47761053650028335, 446350549038171483, 4186889953961917077, 39416115485839527945
OFFSET
1,2
COMMENTS
A related function is theta_3(x) = 1 + 2*Sum_{n>=1} 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} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 8*A(x))^(n-1) = 1 - 6*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 8*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 8*x^n*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 8*x^n*A(x))^(n+1) = 1 - 6*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 8*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 8*x^2 + 63*x^3 + 489*x^4 + 3761*x^5 + 28911*x^6 + 224174*x^7 + 1768801*x^8 + 14298852*x^9 + 118834966*x^10 + 1014912939*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*x - 6*x^4 - 6*x^9 - 6*x^16 - 6*x^25 - 6*x^36 - 6*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06579433445460281447496748523290398966344297589844019028...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 8*A)^n = 4 - 3*Pi^(1/4)/gamma(3/4) = 0.740695566360075956...
(V.2) Let A = A(exp(-2*Pi)) = 0.001895757786183755555448115532175643265455444051246465664...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 8*A)^n = 4 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.98879534353678272...
(V.3) Let A = A(-exp(-Pi)) = -0.03207876150064786089070312769117792591667175850120792604...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 8*A)^n = 4 - 3*(Pi/2)^(1/4)/gamma(3/4) = 1.25926258553164953...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001839948412029108042031275075360099309960919616491079407...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 8*A)^n = 4 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.01120465631727859...
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 - 8*Ser(A))^m ) - 1 + 6*sum(m=1, #A, x^(m^2) ), #A-1)/8 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved