A275425 Number of set partitions of [n] such that seven is a multiple of each block size.

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 5149, 32176, 217361, 1186329, 5282785, 20004037, 66589681, 266164921, 2012163385, 18230119678, 137986473241, 849028203101, 4391743155801, 19722685412431, 98510163677641, 856572597342541, 9516244046786101

Offset: 0

Author

Alois P. Heinz, Jul 27 2016

Keywords

• nonn

Examples

			a(8) = 9: 1234567|8, 1234568|7, 1234578|6, 1234678|5, 1235678|4, 1245678|3, 1345678|2, 1|2345678, 1|2|3|4|5|6|7|8.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..618
• Wikipedia, Partition of a set

Crossrefs

Column k=7 of A275422.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 7]))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 7}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

PARI

a(n) = n!*sum(k=0, n\7, 1/7!^k*binomial(n-6*k, k)/(n-6*k)!); \\ Seiichi Manyama, Feb 26 2022

PARI

a(n) = if(n<7, 1, a(n-1)+binomial(n-1, 6)*a(n-7)); \\ Seiichi Manyama, Feb 26 2022

Formula

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

From Seiichi Manyama, Feb 26 2022: (Start)

a(n) = n! * Sum_{k=0..floor(n/7)} (1/7!)^k * binomial(n-6*k,k)/(n-6*k)!.

a(n) = a(n-1) + binomial(n-1,6) * a(n-7) for n > 6. (End)