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

%I A275425 #14 Feb 26 2022 10:49:24
%S A275425 1,1,1,1,1,1,1,2,9,37,121,331,793,1717,5149,32176,217361,1186329,
%T A275425 5282785,20004037,66589681,266164921,2012163385,18230119678,
%U A275425 137986473241,849028203101,4391743155801,19722685412431,98510163677641,856572597342541,9516244046786101
%N A275425 Number of set partitions of [n] such that seven is a multiple of each block size.
%H A275425 Alois P. Heinz, <a href="/A275425/b275425.txt">Table of n, a(n) for n = 0..618</a>
%H A275425 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275425 E.g.f.: exp(x+x^7/7!).
%F A275425 From _Seiichi Manyama_, Feb 26 2022: (Start)
%F A275425 a(n) = n! * Sum_{k=0..floor(n/7)} (1/7!)^k * binomial(n-6*k,k)/(n-6*k)!.
%F A275425 a(n) = a(n-1) + binomial(n-1,6) * a(n-7) for n > 6. (End)
%e A275425 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.
%p A275425 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A275425       `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 7]))
%p A275425     end:
%p A275425 seq(a(n), n=0..30);
%t A275425 a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 7}}]];
%t A275425 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%o A275425 (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
%o A275425 (PARI) a(n) = if(n<7, 1, a(n-1)+binomial(n-1, 6)*a(n-7)); \\ _Seiichi Manyama_, Feb 26 2022
%Y A275425 Column k=7 of A275422.
%K A275425 nonn
%O A275425 0,8
%A A275425 _Alois P. Heinz_, Jul 27 2016