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A309538
Total number of factorial parts in all compositions of n.
1
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
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
Sequence in context: A298371 A367396 A106472 * A036885 A247300 A137682
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 06 2019
STATUS
approved