OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^(n*(n+1)).
(2) A(x) = sqrt( (1/x) * Series_Reversion( x*eta(x)^2/eta(x^2)^4 ) ), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(3) A(x) = Product_{n>=1} (1 + (x*A(x)^2)^n) * (1 - (x*A(x)^2)^(2*n)).
(4) A(x) = exp( Sum_{n>=1} ( (x*A(x)^2)^n / (1 + (x*A(x)^2)^n) )/n ).
(5) A(x)^4 = Sum_{n>=0} (2*n+1) * (x*A(x)^2)^n / (1 - (x*A(x)^2)^(2*n+1)).
(6) A(x^2)^2 = Sum_{n>=0} (x*A(x^2))^n / (1 + (x*A(x^2))^(2*n+1)).
a(n) ~ c * d^n / n^(3/2), where d = 5.33733388876021052204016376282654316742329168165380444126... and c = 0.24712373554952847890961627688964866920906379264976188659... - Vaclav Kotesovec, Jun 13 2025
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 87*x^5 + 359*x^6 + 1535*x^7 + 6758*x^8 + 30431*x^9 + 139442*x^10 + 648001*x^11 + 3046730*x^12 + ...
where A(x) = 1 + x*A(x)^2 + x^3*A(x)^6 + x^6*A(x)^12 + x^10*A(x)^20 + x^15*A(x)^30 + ... + x^(n*(n+1)/2) * A(x)^(n*(n+1)) + ...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, 2*(s/Sqrt[Pi*(-5 + 8*r^(7/8)*s^(3/4) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[r]*s])])} /. FindRoot[{2*r^(1/8)*s^(5/4) == EllipticTheta[2, 0, Sqrt[r]*s], 5*s^(1/4) == 2*r^(3/8) * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[r]*s]}, {r, 1/4}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 13 2025 *)
PROG
(PARI) {a(n) = my(A = sqrt( (1/x) * serreverse( x*eta(x +x^2*O(x^n))^2/eta(x^2 +x^2*O(x^n))^4 ) ) ); polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 13 2025
STATUS
approved