OFFSET
1,1
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.
Equals the row sums of triangle A354650. - Paul D. Hanna, Jul 27 2022
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) satisfies:
(1) -1 = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-x)^n.
(2) -1 = Sum_{n>=0} A(x)^(n*(n+1)/2) * (1 - x^(2*n+1)) / (-x)^n.
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(n*(n-1)/2) * (-1/x)^n.
(4) 1/x = Product_{n>=1} (1 - A(x)^n) * (1 - A(x)^n*x) * (1 - A(x)^(n-1)/x).
(5) A(x) = Series_Reversion( x*Q(x) ), where Q(x) is the g.f. of A268301 and satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n*Q(x)) * (1 - x^(n-1)/Q(x)).
(6) x = Sum_{n>=1} a(n) * x^n * Q(x)^n, where Q(x) = Sum_{n>=0} A268301(n)/2*(x/4)^n.
a(n) ~ c * d^n / n^(3/2), where d = 24.827421130209954998234265953843191542003179657... and c = 0.020386712793003585674903530668163000681070027... . - Vaclav Kotesovec, Mar 02 2016
EXAMPLE
G.f.: A(x) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...
where A(x) satisfies the Jacobi Triple Product:
-1 = (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) +...
also
1/x = (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) *...
further,
-1 = (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 +...
RELATED SERIES.
The series reversion of g.f. A(x) equals x*Q(x), where Q(x) begins:
Q(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 - 1979248977748/2*x^10/4^10 +...+ A268301(n)/2*x^n/4^n +...
and where Q(x) satisfies the Jacobi Triple Product:
-1 = (1-x)*(1-x*Q(x))*(1-1/Q(x)) * (1-x^2)*(1-x^2*Q(x))*(1-x/Q(x)) * (1-x^3)*(1-x^3*Q(x))*(1-x^2/Q(x)) * (1-x^4)*(1-x^4*Q(x))*(1-x^3/Q(x)) * (1-x^5)*(1-x^5*Q(x))*(1-x^4/Q(x)) * (1-x^6)*(1-x^6*Q(x))*(1-x^5/Q(x)) *...
MATHEMATICA
(* Calculation of constant d: *) 1/r /. FindRoot[{r*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, 1/24}, {s, 1/8}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
PROG
(PARI) {a(n) = my(Q=1/2, t=floor(sqrt(2*n+1)+1/2)); for(i=0, n, Q = (Q + sum(m=-t, t, x^(m*(m-1)/2) * (-Q)^m +x*O(x^n)) )/2 ); polcoeff(serreverse(x*Q), n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 26 2016
STATUS
approved