OFFSET
0,3
COMMENTS
The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..630
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.
(1) x = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 + x^n).
(2) -x*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).
(3) x*theta_4(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).
(4) -x*theta_4(x)*A(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).
(5.a) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(5.b) x*theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) / A(x)^n.
(6.a) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(6.b) -x*theta_4(x)*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
a(n) ~ c * d^n / n^(3/2), where d = 4.083846421711383847604673417919116998017... and c = 0.584432537831593677040363592052688856... - Vaclav Kotesovec, Jun 01 2025
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 49*x^5 + 154*x^6 + 513*x^7 + 1747*x^8 + 6078*x^9 + 21439*x^10 + ...
RELATED SERIES.
1/A(x) = 1 - x - x^2 - 3*x^3 - 5*x^4 - 16*x^5 - 45*x^6 - 155*x^7 - 512*x^8 - 1763*x^9 - 6084*x^10 + ...
SPECIFIC VALUES.
A(exp(-Pi)) = 1.0474973549949421045732567080496722542518531011526934631...
where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n+1)/2) * A(exp(-Pi))^n = exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = 0.03947933420376592813...
A(-exp(-Pi)) = 0.960086060200580366759936974556134222228793624085744940...
where Sum_{n=-oo..+oo} (-1)^(n*(n-1)/2) * exp(-Pi*n*(n+1)/2) * A(-exp(-Pi))^n = -exp(-Pi) * Pi^(1/4) / gamma(3/4) = -0.04694910513068872743...
A(t) = 2 at t = 0.24484187571695922418922496399796775078115821427621282...
A(t) = 7/4 at t = 0.239324355731620083092236573970947000576283799760943...
A(t) = 5/3 at t = 0.234439889083627870257298020352799276294012688627782...
A(t) = 3/2 at t = 0.217134571709901433113197085617818478214816713922905...
A(t) = 4/3 at t = 0.183806911401666173138177455971709388630788740531594...
A(t) = 5/4 at t = 0.157416870441717618165825450612923233287765184975643...
A(1/5) = 1.401449039483961854381757985869052435618161722574956...
A(1/6) = 1.276318946972284528693666572724710434062725174240448...
A(1/7) = 1.213287805382388838362413216213677242108560133326140...
A(1/8) = 1.174388177498186580244775740286834758637341200438483...
A(1/9) = 1.147764942051942680447238410304951699474657455354304...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, -s*Log[r]/2* Sqrt[((s-1)*(-2*r*(s - 1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + QPochhammer[-1, r]^2 * QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1)* QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r])))/ (Pi*QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[ r]^2 + (s-1)^2 * (QPolyGamma[1, -Log[s]/Log[r], r] + QPolyGamma[1, Log[s]/Log[r], r])))]} /. FindRoot[{QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == 2*r*(1 - s), s*Log[r] + (s-1) * (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/s]/Log[r], r]) == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 01 2025 *)
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=2, n, A=concat(A, 0);
A[#A] = polcoef(x - prod(n=1, #A, (1 - x^n*Ser(A)) * (1 - x^(n-1)/Ser(A)) * (1 + x^n) ), #A-1); ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 25 2025
STATUS
approved