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A370037
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
13
1, 7, 48, 323, 2150, 14349, 97431, 681857, 4956116, 37422943, 291763607, 2327820547, 18848767552, 153975563934, 1264733865463, 10431060837749, 86368123241833, 718121985169658, 5997857713743011, 50325664101701349, 424138198629299217, 3589151537280637957, 30481898682409007792
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 - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 7*A(x))^(n-1) = 1 - 5*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 7*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 7*x^n*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 7*x^n*A(x))^(n+1) = 1 - 5*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 7*x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x + 7*x^2 + 48*x^3 + 323*x^4 + 2150*x^5 + 14349*x^6 + 97431*x^7 + 681857*x^8 + 4956116*x^9 + 37422943*x^10 + 291763607*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*x - 5*x^4 - 5*x^9 - 5*x^16 - 5*x^25 - 5*x^36 - 5*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06174306640715063509845961016774795661670689719654375131...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 7*A)^n = (7 - 5*Pi^(1/4)/gamma(3/4))/2 = 0.78391297196672996...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892170701611855386420113452656768397809538392272023405...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 7*A)^n = (7 - 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.99066278628065227...
(V.3) Let A = A(-exp(-Pi)) = -0.03314064170176376172583062314299135400117746157373562359...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 7*A)^n = (7 - 5*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.2160521546097079...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001843340051041967867985717685295689652563446679869985649...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 7*A)^n = (7 - 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.00933721359773216...
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 - 7*Ser(A))^m ) - 1 + 5*sum(m=1, #A, x^(m^2) ), #A-1)/7 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved