A327710 Number of compositions of n into distinct parts such that the difference between any two parts is at least two.

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

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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);
# second Maple program:
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

Cf. A003114, A032020, A268187, A268188, A328222.

Sequence in context: A237714 A245145 A092316 * A339101 A142456 A098508

Adjacent sequences: A327707 A327708 A327709 * A327711 A327712 A327713

Keyword

nonn

Author

Alois P. Heinz, Feb 24 2020

Status

approved