A343667 Number of partitions of an n-set without blocks of size 7.

1, 1, 2, 5, 15, 52, 203, 876, 4132, 21075, 115375, 673620, 4172413, 27296089, 187891174, 1356343385, 10238632307, 80615222404, 660560758879, 5621465069117, 49594663447612, 452846969975391, 4273130715906123, 41612346388251187, 417668648929556073, 4315893703814296053

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

E.g.f.: exp(exp(x) - 1 - x^7/7!).

a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=7, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

Mathematica

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^7/7!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Crossrefs

Cf. A000110, A000296, A027341, A097514, A124504, A343664, A343665, A343666, A343668, A343669.

Sequence in context: A287667 A192855 A148092 * A276723 A287585 A287278

Adjacent sequences: A343664 A343665 A343666 * A343668 A343669 A343670

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 25 2021

Status

approved