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%I #24 Oct 04 2018 09:56:50
%S 1,1,2,4,10,23,61,157,426,1163,3253,9172,26236,75634,220021,644305,
%T 1898977,5626720,16754652,50104781,150427938,453214878,1369857943,
%U 4152559458,12621816592,38459047705,117453028937,359455509767,1102239999454,3386090204843,10419804578693,32115276396739,99131502581481,306422345148052,948423189115351
%N Inverse Euler transform of A195980.
%C The sequence is conjectured to be positive, nondecreasing and strictly convex.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H 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.
%t nmax = 35;
%t theta0[x_, y_] = Sum[x^n y^(n (n-1)/2), {n, 0, (1/2) (1 + Sqrt[1 + 8 nmax]) // Ceiling}];
%t xi0[y_] = -Sum[b[n] y^n, {n, 0, nmax}];
%t cc = CoefficientList[theta0[xi0[y], y] + O[y]^(nmax + 1) // Normal // Collect[#, y]&, y];
%t Do[s[n] = Solve[cc[[n+1]] == 0][[1, 1]]; cc = cc /. s[n], {n, 0, nmax}];
%t A195980 = Table[b[n] /. s[n], {n, 1, nmax}];
%t mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t 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]];
%t EULERi[A195980] (* _Jean-François Alcover_, Oct 04 2018 *)
%Y Cf. A195980, A206000.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Feb 02 2012, Feb 03 2012
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