A357233 a(n) = coefficient of x^n in power series A(x) such that: 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).

1, 1, 3, 11, 46, 207, 980, 4810, 24258, 124951, 654587, 3476985, 18682885, 101372340, 554655435, 3056823864, 16953795008, 94555853982, 529986289496, 2983788539017, 16865736120654, 95677703975144, 544554485912572, 3108656601838926, 17794927199793895

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas, some of which may use A = A(x) for brevity.

(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).

(2) 0 = 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5/(1 - x^2*A^3*(1 - x^3*A^3)/(1 + x^6*A^7/(1 - x^3*A^4*(1 - x^4*A^4)/(1 + ...))))))))), a continued fraction due to an identity of a partial elliptic theta function.

(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A195980. - Paul D. Hanna, Jul 13 2023

Example

G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 207*x^5 + 980*x^6 + 4810*x^7 + 24258*x^8 + 124951*x^9 + 654587*x^10 + 3476985*x^11 + 18682885*x^12 + ...
such that
0 = 1 - A(x) + x*A(x)^3 - x^3*A(x)^6 + x^6*A(x)^10 - x^10*A(x)^15 + x^15*A(x)^21 - x^21*A(x)^28 + ... + (-1)^n*x^(n*(n-1)/2)*A(x)^(n*(n+1)/2) + ...
SPECIFIC VALUES.
A(1/7) = 1.2997111125331190764482142994969231...
A(1/8) = 1.221202992288263902503896694281250380662689...
CONTINUED FRACTION.
The continued fraction in formula (2) may be seen to converge to zero as a limit of successive steps that begin as follows:
[2] 1/(1 + A/(1 - A*(1 - x*A)))
[3] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3)))
[4] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)))))
[5] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5)))))
[6] 1/(1 + A/(1 - A*(1 - x*A)/(1 + x^2*A^3/(1 - x*A^2*(1 - x^2*A^2)/(1 + x^4*A^5/(1 - x^2*A^3*(1 - x^3*A^3)))))))
...
substituting A = A(x), the resulting power series in x are:
[2] x^2 - 3*x^3 - 13*x^4 - 58*x^5 - 275*x^6 - 1350*x^7 + ...
[3] x^3 - 5*x^4 - 23*x^5 - 111*x^6 - 553*x^7 - 2820*x^8 + ...
[4] x^7 + 11*x^8 + 87*x^9 + 602*x^10 + 3894*x^11 + 24245*x^12 + ...
[5] x^9 + 14*x^10 + 132*x^11 + 1046*x^12 + 7538*x^13 + ...
[6] -x^15 - 21*x^16 - 273*x^17 - 2821*x^18 - 25432*x^19 + ...
[7] -x^18 - 25*x^19 - 375*x^20 - 4375*x^21 - 43800*x^22 + ...
[8] x^26 + 34*x^27 + 663*x^28 + 9725*x^29 + 119226*x^30 + ...
...
the limit of these series converges to zero for |x| < r < 1 where r is the radius of convergence of g.f. A(x).

Prog

(PARI) {a(n) = my(A=[1], M=1); for(i=1, n, A = concat(A, 0); M = ceil(sqrt(2*(#A)+1));
A[#A] = polcoeff( sum(n=0, M, (-1)^n * x^(n*(n-1)/2) * Ser(A)^(n*(n+1)/2) ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A195980, A193111, A107590.

Sequence in context: A291090 A155134 A006605 * A306822 A193074 A287891

Adjacent sequences: A357230 A357231 A357232 * A357234 A357235 A357236

Keyword

nonn

Author

Paul D. Hanna, Oct 17 2022

Status

approved