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

1,2

Links

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

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

Example

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, ...

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 *)

Crossrefs

Cf. A248686.

Sequence in context: A318372 A157313 A030971 * A006932 A001040 A181949

Adjacent sequences: A248684 A248685 A248686 * A248688 A248689 A248690

Keyword

nonn,easy

Author

Clark Kimberling, Oct 11 2014

Status

approved