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A226999 Inverse Euler transform of A005169 (fountains of coins).

%I A226999 #28 Dec 15 2024 04:37:18
%S A226999 1,0,1,1,2,3,5,8,13,21,35,55,93,149,248,403,671,1098,1827,3013,5013,
%T A226999 8313,13859,23063,38534,64341,107715,180355,302565,507784,853507,
%U A226999 1435415,2416941,4072272,6868062,11590807,19577555,33088481,55964327,94712212
%N A226999 Inverse Euler transform of A005169 (fountains of coins).
%C A226999 If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.
%D A226999 Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.
%H A226999 Alois P. Heinz, <a href="/A226999/b226999.txt">Table of n, a(n) for n = 1..4178</a>
%H A226999 R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H A226999 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A226999 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A226999 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%F A226999 a(n) ~ 1 / (n * r^n), where r = A347901 = 0.57614876914275660229786857371993878235472466311897446868515653431946822937499... - _Vaclav Kotesovec_, Oct 09 2019
%t A226999 max = 100;
%t A226999 A005169 = Series[1 - Fold[Function[1 - x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]&;
%t A226999 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A226999 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 A226999 EULERi[A005169 // Rest] (* _Jean-François Alcover_, Jan 06 2020 *)
%Y A226999 Cf. A005169, A005170.
%K A226999 nonn
%O A226999 1,5
%A A226999 _R. J. Mathar_, Jun 26 2013