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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}.

%I A195980 #46 Jul 20 2023 10:05:19
%S A195980 1,1,2,4,9,21,52,133,351,948,2610,7298,20672,59192,171059,498275,
%T A195980 1461437,4312300,12792342,38128354,114126797,342914278,1033914760,
%U A195980 3127154610,9485523742,28848101993,87948036401,268724650863,822791384597,2524113596369,7757247543181,23880003051017,73627904162143,227347168628991,702970760225573,2176459051318522
%N 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}.
%C A195980 Sokal (2011) shows that all the terms are positive.
%H A195980 Paul D. Hanna, <a href="/A195980/b195980.txt">Table of n, a(n) for n = 0..400</a>
%H A195980 Ramón Flores and Juan González-Meneses, <a href="https://arxiv.org/abs/1808.03066">On the growth of Artin--Tits monoids and the partial theta function</a>, arXiv:1808.03066 [math.GR], 2018.
%H A195980 Thomas Prellberg, <a href="https://arxiv.org/abs/1210.0095">The combinatorics of the leading root of the partial theta function</a>, arXiv:1210.0095 [math.CO], 2012.
%H A195980 A. D. Sokal, <a href="http://arxiv.org/abs/1106.1003">The leading root of the partial theta function</a>, arXiv preprint arXiv:1106.1003 [math.CO], 2011-2012.  Adv. Math. 229 (2012), no. 5, 2603-2621.
%H A195980 A. D. Sokal, <a href="http://arxiv.org/src/1106.1003v1/anc/partialtheta_xi0.m">The first 6999 terms</a>
%F A195980 From _Paul D. Hanna_, Jul 13 2023: (Start)
%F A195980 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
%F A195980 (1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^n (see Sokal references).
%F A195980 (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)
%e A195980 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 + ...
%e A195980 such that
%e A195980 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 + ...
%t A195980 nmax = 35;
%t A195980 theta0[x_, y_] = Sum[x^n y^(n(n-1)/2), {n, 0, (1/2)(1+Sqrt[1+8nmax]) // Ceiling}];
%t A195980 xi0[y_] = -Sum[a[n] y^n, {n, 0, nmax}];
%t A195980 cc = CoefficientList[theta0[xi0[y], y] + O[y]^(nmax+1) // Normal // Collect[#,y]&, y];
%t A195980 Do[s[n] = Solve[cc[[n+1]] == 0][[1, 1]]; cc = cc /. s[n] , {n, 0, nmax}];
%t A195980 Table[a[n] /. s[n], {n, 0, nmax}] (* _Jean-François Alcover_, Sep 05 2018 *)
%Y A195980 Cf. A195981, A195982, A205999, A206000, A357233.
%K A195980 nonn
%O A195980 0,3
%A A195980 _N. J. A. Sloane_, Sep 25 2011, Feb 01 2012