OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
a(6) = 53 = 6+4+5+5+3+3+6+4+6+5+6: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
MAPLE
b:= proc(n, i, k, m) option remember; `if`(i<k or n>
(2*i-k+1)*k/2, 0, `if`(n=0, [1, 0], b(n, i-1, k, m)+
(p-> p+[0, p[1]*i/(m+1-k)])(b(n-i, min(n-i, i-1), k-1, m))))
end:
a:= n-> add(b(n$2, k$2)[2]*k!, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, i_, k_, m_] := b[n, i, k, m] = If[i < k || n >
(2*i - k + 1)*k/2, {0, 0}, If[n == 0, {1, 0}, b[n, i - 1, k, m] +
Function[p, p+{0, p[[1]]*i/(m+1-k)}][b[n-i, Min[n-i, i-1], k-1, m]]]];
a[n_] := Sum[b[n, n, k, k][[2]]*k!, {k, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 04 2020
STATUS
approved