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A327710
Number of compositions of n into distinct parts such that the difference between any two parts is at least two.
3
1, 1, 1, 1, 3, 3, 5, 5, 7, 13, 15, 21, 29, 35, 43, 55, 87, 99, 137, 173, 235, 277, 363, 429, 545, 755, 895, 1135, 1443, 1827, 2285, 2837, 3463, 4285, 5199, 6309, 8237, 9755, 12091, 14743, 18351, 22251, 27833, 33125, 40819, 49045, 59691, 70869, 86033, 106163
OFFSET
0,5
COMMENTS
All terms are odd.
LINKS
FORMULA
a(n) = Sum_{k>=0} k! * A268187(n,k).
G.f.: Sum_{k>=0} k! * x^(k^2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Dec 04 2020
EXAMPLE
a(9) = 13: 135, 153, 315, 351, 513, 531, 36, 63, 27, 72, 18, 81, 9.
MAPLE
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, p!, b(n, i-1, p)+b(n-i, min(n-i, i-2), p+1)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);
# Alternative:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(k!*b(n-k^2, k), k=0..floor(sqrt(n))):
seq(a(n), n=0..50);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[k!*b[n - k^2, k], {k, 0, Floor[Sqrt[n]]}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 24 2020
STATUS
approved