A331518 a(n) = Sum_{k=0..n} q(n,k) * !k, where q(n,k) = number of partitions of n into k distinct parts and !k = subfactorial of k.
1, 0, 0, 1, 1, 2, 4, 5, 7, 10, 21, 24, 37, 49, 71, 129, 160, 227, 313, 433, 572, 1012, 1213, 1750, 2315, 3223, 4159, 5740, 8945, 11206, 15402, 20506, 27545, 36068, 48122, 61960, 94694, 116240, 158580, 205397, 276458, 352526, 470101, 596433, 781224, 1111228
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Programs
-
Maple
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end: b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m+1))) end: a:= n-> b(n$2, 0): seq(a(n), n=0..45); # Alois P. Heinz, Mar 02 2024 -
Mathematica
Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] Subfactorial[k], {k, 0, n}], {n, 0, 45}] nmax = 45; CoefficientList[Series[Sum[Subfactorial[k] x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] nmax = 50; CoefficientList[Series[Sum[Subfactorial[k] * x^(k*(k+1)/2) / Product[(1 - x^j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2020 *)
Formula
G.f.: Sum_{k>=0} !k * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j).
Comments