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%I #28 Nov 23 2023 20:52:13
%S 1,1,2,4,10,20,58,124,344,811,2071,4973,15454,36031,96212,237563,
%T 668695,1626751,4674373,11470722,31460456,81705943,224598113
%N Number of set partitions of [5n] into 5-element subsets {i, i+k, i+2k, i+3k, i+4k} with 1<=k<=n.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e a(4) = 10: {{1,2,3,4,5}, {6,7,8,9,10}, {11,12,13,14,15}, {16,17,18,19,20}},
%e {{1,3,5,7,9}, {2,4,6,8,10}, {11,12,13,14,15}, {16,17,18,19,20}},
%e {{1,2,3,4,5}, {6,8,10,12,14}, {7,9,11,13,15}, {16,17,18,19,20}},
%e {{1,4,7,10,13}, {2,5,8,11,14}, {3,6,9,12,15}, {16,17,18,19,20}},
%e {{1,2,3,4,5}, {6,7,8,9,10}, {11,13,15,17,19}, {12,14,16,18,20}},
%e {{1,3,5,7,9}, {2,4,6,8,10}, {11,13,15,17,19}, {12,14,16,18,20}},
%e {{1,5,9,13,17}, {2,4,6,8,10}, {3,7,11,15,19}, {12,14,16,18,20}},
%e {{1,2,3,4,5}, {6,9,12,15,18}, {7,10,13,16,19}, {8,11,14,17,20}},
%e {{1,3,5,7,9}, {2,6,10,14,18}, {4,8,12,16,20}, {11,13,15,17,19}},
%e {{1,5,9,13,17}, {2,6,10,14,18}, {3,7,11,15,19}, {4,8,12,16,20}}.
%p b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
%p `if`({seq(m-h*j, h=1..4)} minus s={}, b(s minus {seq(m-h*j,
%p h=0..4)}, t), 0), j=1..min(t, iquo(m-1, 4))))(max(s)))
%p end:
%p a:= proc(n) option remember; forget(b): b({$1..5*n}, n) end:
%p seq(a(n), n=0..10);
%t b[s_, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[Union[Table[m - h*j, {h, 1, 4}] ~Complement~ s] == {}, b[s ~Complement~ Union[Table[m - h*j, {h, 0, 4}]], t], 0], {j, 1, Min[t, Quotient[m-1, 4]]}]][Max[s]]];
%t a[n_] := a[n] = b[Range[5n], n];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)
%Y Cf. A000567 (number of subsets), A008587 (number of elements), A104431 (when k is unbounded), A337520.
%Y Main diagonal of A360491.
%K nonn,more
%O 0,3
%A _Alois P. Heinz_, Nov 17 2021
%E a(22) from _Alois P. Heinz_, Nov 23 2022
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