A317807 Number of set partitions of [k] into 5 blocks with equal element sum, where k is the n-th positive integer that allows such a partition.

1, 1, 68, 187, 27763, 108516, 25958279, 100664383, 26388943467, 109026138857, 33100108402861, 139752234469078, 46498731704890104, 200612215343574676, 71799817534098086846, 314741192906319529056

Offset: 1

Author

Alois P. Heinz, Aug 07 2018

Keywords

• nonn

Comments

k = 9, 10, 14, 15, 19, ... A047208(n+3) for n = 1, 2, 3, 4, 5, ... .

Examples

			a(1) = 1: 18|27|36|45|9 with k = 9.
a(2) = 1: 1(10)|29|38|47|56 with k = 10.
		

Links

• Wikipedia, Partition of a set

Crossrefs

Cf. A047208, A275714.

Programs

Maple

b:= proc() option remember; local i, j, t; `if`(args[1]=0,
      `if`(nargs=2, 1, b(args[t] $t=2..nargs)), add(
      `if`(args[j] -args[nargs]<0, 0, b(sort([seq(args[i]-
      `if`(i=j, args[nargs], 0), i=1..nargs-1)])[],
                args[nargs]-1)), j=1..nargs-1))
    end:
a:= proc(n) option remember; (k-> (m->
      b((m/5)$5, k)/5!)(k*(k+1)/2))(5+5*n/2+3/4*(1-(-1)^n))
    end:
seq(a(n), n=1..8);

Mathematica

b[args_List] := b[args] = Module[{nargs = Length[args]}, If[args[[1]] == 0, If[nargs == 3, 1, b[args // Rest]], Sum[If[args[[j]] - Last[args] < 0, 0, b[Append[Sort[Flatten[Table[args[[i]] - If[i == j, Last[args], 0], {i, 1, nargs - 1}]]], Last[args] - 1]]], {j, 1, nargs - 1}]]];
a[n_] := a[n] = Function[k, Function[m, b[Append[Table[m/5, {5}], k]]/5!][k (k + 1)/2]][5 + 5n/2 + (3/4)(1 - (-1)^n)];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 12}] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Formula

a(n) = A275714(A047208(n+3),5).