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 F(x) satisfy the following formulas.
(1.a) F(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)).
(1.b) 1/F(x) = Sum_{n>=0} (-1)^n * x^n * A(x)^(n^2).
(2.a) F(x) = 1/(1 - x/(1 - (A^2-1)*x/(1 - A^4*x/(1 - (A^6-A^2)*x/(1 - A^8*x/(1 - (A^10-A^4)*x/(1 - A^12*x/(1 - (A^14-A^6)*x/(1 - A^16*x/(1 - ...)))))))))), a continued fraction due to an identity of a partial elliptic theta function.
(2.b) F(x) = 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 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 A383558.
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 219*x^4 + 1694*x^5 + 13994*x^6 + 121410*x^7 + 1093480*x^8 + 10141721*x^9 + 96319038*x^10 + ...
RELATED SERIES.
F(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)) = 1 + x + x^2 + 3*x^3 + 18*x^4 + 130*x^5 + 1028*x^6 + 8643*x^7 + 76030*x^8 + 692547*x^9 + 6485384*x^10 + ...
1/F(x) = Sum_{n>=0} (-1)^n * x^n * A(x)^(n^2) = 1 - x - 2*x^3 - 13*x^4 - 97*x^5 - 782*x^6 - 6661*x^7 - 59159*x^8 - 542859*x^9 - 5113500*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*(n-1)) ) - 1/sum(n=0, #V, (-1)^n * x^n * A^(n^2) ), #V) ); V[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2025
STATUS
approved