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A336273
Number of compositions of n containing at least one part p of multiplicity p.
2
0, 1, 0, 2, 3, 5, 14, 28, 44, 101, 207, 399, 779, 1609, 3122, 6121, 11804, 23631, 46273, 91604, 178096, 352419, 691996, 1371306, 2702206, 5356324, 10604748, 21080216, 41869930, 83383786, 166114046, 331434088, 661685588, 1322042390, 2642367028, 5283397304
OFFSET
0,4
LINKS
FORMULA
a(n) = A011782(n) - A336269(n).
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
add(`if`(i=j, 0, b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
end:
a:= n-> ceil(2^(n-1))-b(n$2, 0):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
Sum[If[i == j, 0, b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := Ceiling[2^(n - 1)] - b[n, n, 0];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A283448 A177901 A143743 * A104870 A232162 A243555
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 15 2020
STATUS
approved