A205999 Inverse Euler transform of A195980.

1, 1, 2, 4, 10, 23, 61, 157, 426, 1163, 3253, 9172, 26236, 75634, 220021, 644305, 1898977, 5626720, 16754652, 50104781, 150427938, 453214878, 1369857943, 4152559458, 12621816592, 38459047705, 117453028937, 359455509767, 1102239999454, 3386090204843, 10419804578693, 32115276396739, 99131502581481, 306422345148052, 948423189115351

Offset: 0

Author

N. J. A. Sloane, Feb 02 2012, Feb 03 2012

Keywords

• nonn

Comments

The sequence is conjectured to be positive, nondecreasing and strictly convex.

Links

• N. J. A. Sloane, Transforms
• 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.

Crossrefs

Cf. A195980, A206000.

Programs

Mathematica

nmax = 35;
theta0[x_, y_] = Sum[x^n y^(n (n-1)/2), {n, 0, (1/2) (1 + Sqrt[1 + 8 nmax]) // Ceiling}];
xi0[y_] = -Sum[b[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}];
A195980 = Table[b[n] /. s[n], {n, 1, nmax}];
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i b[[i]] - Sum[c[[d]] b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i) Sum[mob[i, d] c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A195980] (* Jean-François Alcover, Oct 04 2018 *)