OFFSET
1,3
COMMENTS
The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..512
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-1/x)^n.
(2) -x = Sum_{n>=0} A(x)^(n*(n+1)/2) * (1 - x^(2*n+1)) / (-x)^n.
(3) -x = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-x)^n.
(4) -x = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n/x) * (1 - A(x)^(n-1)*x).
(5) -x/(1-x) = Product_{n>=1} (1 - A(x)^n) * (1 - (x+1/x)*A(x)^n + A(x)^(2*n)).
a(n) ~ c * d^n / n^(3/2), where d = 6.1842071022304098678015128954668969... and c = 0.0509064807103441056564968325417718... . - Vaclav Kotesovec, Mar 05 2016
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 50*x^5 + 228*x^6 + 1093*x^7 + 5439*x^8 + 27816*x^9 + 145310*x^10 + 772109*x^11 + 4159998*x^12 + 22674120*x^13 + 124800022*x^14 + 692686326*x^15 + 3872659052*x^16 +...
where A(x) satisfies the Jacobi Triple Product:
1 = (1-A(x))*(1-A(x)*x)*(1-1/x) * (1-A(x)^2)*(1-A(x)^2*x)*(1-A(x)/x) * (1-A(x)^3)*(1-A(x)^3*x)*(1-A(x)^2/x) * (1-A(x)^4)*(1-A(x)^4*x)*(1-A(x)^3/x) * (1-A(x)^5)*(1-A(x)^5*x)*(1-A(x)^4/x) * (1-A(x)^6)*(1-A(x)^6*x)*(1-A(x)^5/x) *...
also
-x = (1-A(x))*(1-A(x)/x)*(1-x) * (1-A(x)^2)*(1-A(x)^2/x)*(1-A(x)*x) * (1-A(x)^3)*(1-A(x)^3/x)*(1-A(x)^2*x) * (1-A(x)^4)*(1-A(x)^4/x)*(1-A(x)^3*x) * (1-A(x)^5)*(1-A(x)^5/x)*(1-A(x)^4*x) * (1-A(x)^6)*(1-A(x)^6/x)*(1-A(x)^5*x) +...
further,
-x = (1-x) - A(x)*(1-x^3)/x + A(x)^3*(1-x^5)/x^2 - A(x)^6*(1-x^7)/x^3 + A(x)^10*(1-x^9)/x^4 - A(x)^15*(1-x^11)/x^5 + A(x)^21*(1-x^13)/x^6 +...
MATHEMATICA
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[1/r, s] * QPochhammer[r, s] * QPochhammer[s, s] == 1 - r, (Log[1-s] + QPolyGamma[0, 1, s]) / (s*Log[s]) - Derivative[0, 1][QPochhammer][1/r, s]/QPochhammer[1/r, s] - Derivative[0, 1][QPochhammer][r, s]/QPochhammer[r, s] - Derivative[0, 1][QPochhammer][s, s]/ QPochhammer[s, s] == 0}, {r, 1/6}, {s, 1/4}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec( sum(m=1, sqrtint(2*#A)+2, (-1)^m*(x*Ser(A))^(m*(m-1)/2)*(1-x^(2*m-1))/x^m) )[#A-1] ); A[n]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 02 2016
STATUS
approved