A309538 Total number of factorial parts in all compositions of n.

0, 1, 3, 7, 17, 40, 93, 210, 469, 1036, 2268, 4928, 10640, 22848, 48832, 103936, 220416, 465920, 982016, 2064384, 4329472, 9060352, 18923520, 39452672, 82116609, 170655746, 354156549, 734003212, 1519386652, 3141533760, 6488588432, 13388218688, 27598521024

Offset: 0

Author

Alois P. Heinz, Aug 06 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3312

Crossrefs

Cf. A000142, A102291.

Programs

Maple

g:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; g(n):=%
    end:
a:= proc(n) option remember; add(a(n-j)+
      `if`(g(j)=1, ceil(2^(n-j-1)), 0), j=1..n)
    end:
seq(a(n), n=0..33);

Mathematica

g[n_] := g[n] = Module[{i, p = 1}, For[i = 2, True, i++, If[n == p, p = 1; Break[], If[nJean-François Alcover, Jan 10 2023, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=1} x^(k!)*(1-x)^2/(1-2*x)^2.

a(n) ~ c * 2^n * n, where c = 0.1914062649011611938476562500000000001880790961... - Vaclav Kotesovec, Aug 18 2019