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

0,3

Links

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

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

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[n<p, p = 0; Break[]]]; p = p*i]; p];
a[n_] := a[n] = Sum[a[n-j] + If[g[j] == 1, Ceiling[2^(n-j-1)], 0], {j, 1, n}];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 10 2023, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A102291.

Sequence in context: A298371 A367396 A106472 * A036885 A247300 A137682

Adjacent sequences: A309535 A309536 A309537 * A309539 A309540 A309541

Keyword

nonn

Author

Alois P. Heinz, Aug 06 2019

Status

approved