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

0,8

Links

Alois P. Heinz, Table of n, a(n) for n = 0..618

Wikipedia, Partition of a set

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)

Example

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.

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 *)

Prog

(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

Crossrefs

Column k=7 of A275422.

Sequence in context: A280351 A306721 A306852 * A212386 A333883 A206374

Adjacent sequences: A275422 A275423 A275424 * A275426 A275427 A275428

Keyword

nonn

Author

Alois P. Heinz, Jul 27 2016

Status

approved