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A370043
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
13
1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, 871650208, 8884142855, 92061370003, 969550433086, 10363557226896, 112215017274331, 1228207449471086, 13561137797537413, 150791851996365182, 1686274213530482843, 18945675318778308411, 213704510012147008821
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 - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 10*A(x))^(n-1) = 1 - 8*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 10*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 10*x^n*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 10*x^n*A(x))^(n+1) = 1 - 8*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 10*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 10*x^2 + 99*x^3 + 971*x^4 + 9461*x^5 + 91959*x^6 + 895518*x^7 + 8775161*x^8 + 86870264*x^9 + 871650208*x^10 + 8884142855*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 8*x - 8*x^4 - 8*x^9 - 8*x^16 - 8*x^25 - 8*x^36 - 8*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07572861892245027340976642864415638836692678958408803376...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 10*A)^n = 5 - 4*Pi^(1/4)/gamma(3/4) = 0.6542607551467679416987...
(V.2) Let A = A(exp(-2*Pi)) = 0.001902972911784356118532074933211699956337964100195554269...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 10*A)^n = 5 - 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9850604580490436358...
(V.3) Let A = A(-exp(-Pi)) = -0.03014664142938059660934561948726688645121488051083843222...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 10*A)^n = 5 - 4*(Pi/2)^(1/4)/gamma(3/4) = 1.34568344737553271437...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001833202439114209450155973975718938793478260093149995057...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 10*A)^n = 5 - 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.0149395417563714568...
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 - 10*Ser(A))^m ) - 1 + 8*sum(m=1, #A, x^(m^2) ), #A-1)/10 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved