OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..302
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n, where A = A(x) along with related series G(x) satisfy the following formulas.
(1.a) G(x) = Sum_{n>=0} x^n * A(x)^(n^2).
(1.b) 1/G(x) = Sum_{n>=0} (-1)^n * x^n * A(x)^(n*(n+1)).
(2.a) G(x) = 1/(1 - A*x/(1 - (A^3-A)*x/(1 - A^5*x/(1 - (A^7-A^3)*x/(1 - A^9*x/(1 - (A^11-A^5)*x/(1 - A^13*x/(1 - (A^15-A^7)*x/(1 - A^17*x/(1 - ...)))))))))), a continued fraction due to an identity of a partial elliptic theta function.
(2.b) G(x) = 1 + A^2*x/(1 + (A^4-A^2)*x/(1 + A^6*x/(1 + (A^8-A^4)*x/(1 + A^10*x/(1 + (A^12-A^6)*x/(1 + A^14*x/(1 + (A^16-A^8)*x/(1 + ...)))))))), a continued fraction due to an identity of a partial elliptic theta function.
(3) A(x) = B(x*A(x)) where B(x) = A(x/B(x)) is the g.f. of A383557.
EXAMPLE
G.f.: A(x) = 1 + x + 6*x^2 + 47*x^3 + 424*x^4 + 4175*x^5 + 43617*x^6 + 475457*x^7 + 5350757*x^8 + 61727826*x^9 + 726360262*x^10 + ...
RELATED SERIES.
G(x) = Sum_{n>=0} x^n * A(x)^(n^2) = 1 + x + 2*x^2 + 11*x^3 + 87*x^4 + 795*x^5 + 7905*x^6 + 83182*x^7 + 911659*x^8 + 10302675*x^9 + 119249502*x^10 + ...
1/G(x) = Sum_{n>=0} (-1)^n * x^n * A(x)^(n*(n+1)) = 1 - x - x^2 - 8*x^3 - 66*x^4 - 615*x^5 - 6188*x^6 - 65642*x^7 - 723687*x^8 - 8215580*x^9 - 95435685*x^10 - ...
PROG
(PARI) {a(n) = my(A, V=[1, 1]); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(n=0, #V, x^n * A^(n^2) ) - 1/sum(n=0, #V, (-1)^n * x^n * A^(n*(n+1)) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2025
STATUS
approved