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

%I A248687 #16 Feb 20 2024 20:29:18
%S A248687 1,3,10,43,221,1371,9696,78751,712447,7173853,79106413,952587175,
%T A248687 12397677007,173864946685,2609479384942,41786786069887,
%U A248687 710577455524223,12795789975272877,243154034699436147,4864103085730989101,102153340062463300261,2247608818115460466681
%N A248687 Sum of the numbers in row n of the triangular array at A248686.
%H A248687 Alois P. Heinz, <a href="/A248687/b248687.txt">Table of n, a(n) for n = 1..450</a> (first 100 terms from Clark Kimberling)
%F A248687 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.
%F A248687 a(n) ~ 2 * n!. - _Vaclav Kotesovec_, Oct 21 2014
%F A248687 a(n) mod 2 = 0 <=> n in { A126646 } \ { 1 }. - _Alois P. Heinz_, Feb 20 2024
%e A248687 First seven rows of the array at A248686:
%e A248687 1
%e A248687 1   2
%e A248687 1   3    6
%e A248687 1   6    12    24
%e A248687 1   10   30    60    120
%e A248687 1   20   90    180   360    720
%e A248687 1   35   210   630   1260   2520   5040
%e A248687 The row sums are 1, 3, 10, ...
%p A248687 b:= proc(n, k) option remember; `if`(k<1,
%p A248687      `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
%p A248687     end:
%p A248687 a:= n-> add(b(n,k), k=0..n):
%p A248687 seq(a(n), n=1..22);  # _Alois P. Heinz_, Feb 20 2024
%t A248687 f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
%t A248687 t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
%t A248687 u = Flatten[t]  (* A248686 sequence *)
%t A248687 TableForm[t]    (* A248686 array *)
%t A248687 Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)
%Y A248687 Cf. A248686.
%K A248687 nonn,easy
%O A248687 1,2
%A A248687 _Clark Kimberling_, Oct 11 2014