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

1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 35, 55, 93, 149, 248, 403, 671, 1098, 1827, 3013, 5013, 8313, 13859, 23063, 38534, 64341, 107715, 180355, 302565, 507784, 853507, 1435415, 2416941, 4072272, 6868062, 11590807, 19577555, 33088481, 55964327, 94712212

Offset

1,5

Comments

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.

References

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..4178

R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.

R. K. Guy, Letter to N. J. A. Sloane, 1987

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]

Formula

a(n) ~ 1 / (n * r^n), where r = A347901 = 0.57614876914275660229786857371993878235472466311897446868515653431946822937499... - Vaclav Kotesovec, Oct 09 2019

Mathematica

max = 100;
A005169 = Series[1 - Fold[Function[1 - x^#2/#1], 1, Range[max, 0, -1]], {x, 0, max}] // CoefficientList[#, x]&;
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[A005169 // Rest] (* Jean-François Alcover, Jan 06 2020 *)

Crossrefs

Cf. A005169, A005170.

Sequence in context: A050762 A321656 A005170 * A218032 A229194 A304790

Adjacent sequences: A226996 A226997 A226998 * A227000 A227001 A227002

Keyword

nonn

Author

R. J. Mathar, Jun 26 2013

Status

approved