A248687 Sum of the numbers in row n of the triangular array at A248686.

1, 3, 10, 43, 221, 1371, 9696, 78751, 712447, 7173853, 79106413, 952587175, 12397677007, 173864946685, 2609479384942, 41786786069887, 710577455524223, 12795789975272877, 243154034699436147, 4864103085730989101, 102153340062463300261, 2247608818115460466681

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• easy

Examples

			First seven rows of the array at A248686:
1
1   2
1   3    6
1   6    12    24
1   10   30    60    120
1   20   90    180   360    720
1   35   210   630   1260   2520   5040
The row sums are 1, 3, 10, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Clark Kimberling)

Crossrefs

Cf. A248686.

Programs

Maple

b:= proc(n, k) option remember; `if`(k<1,
     `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
    end:
a:= n-> add(b(n,k), k=0..n):
seq(a(n), n=1..22);  # Alois P. Heinz, Feb 20 2024

Mathematica

f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A248686 sequence *)
TableForm[t]    (* A248686 array *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)

Formula

a(n) = Sum_{k=1..n} n!/(n(1)!*n(2)!* ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1..k.

a(n) ~ 2 * n!. - Vaclav Kotesovec, Oct 21 2014

a(n) mod 2 = 0 <=> n in { A126646 } \ { 1 }. - Alois P. Heinz, Feb 20 2024