A195980 Coefficients of expansion of "leading root" xi_0(y) of the partial theta function Sum_{n=0..oo} x^n y^{n(n-1)/2}.
1, 1, 2, 4, 9, 21, 52, 133, 351, 948, 2610, 7298, 20672, 59192, 171059, 498275, 1461437, 4312300, 12792342, 38128354, 114126797, 342914278, 1033914760, 3127154610, 9485523742, 28848101993, 87948036401, 268724650863, 822791384597, 2524113596369, 7757247543181, 23880003051017, 73627904162143, 227347168628991, 702970760225573, 2176459051318522
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 52*x^6 + 133*x^7 + 351*x^8 + 948*x^9 + 2610*x^10 + 7298*x^11 + 20672*x^12 + ... such that 0 = 1 - A(x) + x*A(x)^2 - x^3*A(x)^3 + x^6*A(x)^4 - x^10*A(x)^5 + x^15*A(x)^6 - x^21*A(x)^7 + ... + (-1)^n*x^(n*(n-1)/2)*A(x)^n + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..400
- Ramón Flores and Juan González-Meneses, On the growth of Artin--Tits monoids and the partial theta function, arXiv:1808.03066 [math.GR], 2018.
- Thomas Prellberg, The combinatorics of the leading root of the partial theta function, arXiv:1210.0095 [math.CO], 2012.
- A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011-2012. Adv. Math. 229 (2012), no. 5, 2603-2621.
- A. D. Sokal, The first 6999 terms
Programs
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Mathematica
nmax = 35; theta0[x_, y_] = Sum[x^n y^(n(n-1)/2), {n, 0, (1/2)(1+Sqrt[1+8nmax]) // Ceiling}]; xi0[y_] = -Sum[a[n] y^n, {n, 0, nmax}]; cc = CoefficientList[theta0[xi0[y], y] + O[y]^(nmax+1) // Normal // Collect[#,y]&, y]; Do[s[n] = Solve[cc[[n+1]] == 0][[1, 1]]; cc = cc /. s[n] , {n, 0, nmax}]; Table[a[n] /. s[n], {n, 0, nmax}] (* Jean-François Alcover, Sep 05 2018 *)
Formula
From Paul D. Hanna, Jul 13 2023: (Start)
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^n (see Sokal references).
(2) 0 = 1/(1 + A(x)/(1 - A(x)*(1 - x)/(1 + x^2*A(x)/(1 - x*A(x)*(1 - x^2)/(1 + x^4*A(x)/(1 - x^2*A(x)*(1 - x^3)/(1 + x^6*A(x)/(1 - x^3*A(x)*(1 - x^4)/(1 + ...))))))))), a continued fraction due to an identity of a partial elliptic theta function. (End)
Comments