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A275423 Number of set partitions of [n] such that five is a multiple of each block size.

%I A275423 #32 Nov 19 2024 02:53:07
%S A275423 1,1,1,1,1,2,7,22,57,127,379,1849,9109,37324,128129,507508,3031393,
%T A275423 19609773,108440893,500515633,2467616641,17154715726,134519207131,
%U A275423 927764339426,5359830269641,31580724696907,248587878630807,2259650025239257,18541914182165557
%N A275423 Number of set partitions of [n] such that five is a multiple of each block size.
%H A275423 Alois P. Heinz, <a href="/A275423/b275423.txt">Table of n, a(n) for n = 0..619</a>
%H A275423 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A275423 E.g.f.: exp(x+x^5/5!).
%F A275423 From _Seiichi Manyama_, Feb 26 2022: (Start)
%F A275423 a(n) = n! * Sum_{k=0..floor(n/5)} (1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.
%F A275423 a(n) = a(n-1) + binomial(n-1,4) * a(n-5) for n > 4. (End)
%F A275423 a(n) = hypergeom([-n/5,(1-n)/5,(2-n)/5,(3-n)/5,(4-n)/5],[],-625/24). - _Karol A. Penson_, Sep 14 2023.
%e A275423 a(6) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
%p A275423 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A275423       `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 5]))
%p A275423     end:
%p A275423 seq(a(n), n=0..30);
%p A275423 # second Maple program:
%p A275423 seq(simplify(hypergeom([-n/5, (1-n)/5, (2-n)/5, (3-n)/5, (4-n)/5], [], -625/24)), n = 0..28); # _Karol A. Penson_, Sep 14 2023.
%t A275423 a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 5}}]];
%t A275423 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%o A275423 (PARI) a(n) = n!*sum(k=0, n\5, 1/5!^k*binomial(n-4*k, k)/(n-4*k)!); \\ _Seiichi Manyama_, Feb 26 2022
%o A275423 (PARI) a(n) = if(n<5, 1, a(n-1)+binomial(n-1, 4)*a(n-5)); \\ _Seiichi Manyama_, Feb 26 2022
%Y A275423 Column k=5 of A275422.
%K A275423 nonn
%O A275423 0,6
%A A275423 _Alois P. Heinz_, Jul 27 2016