A193463 Row sums of triangle A076732.

1, 1, 6, 22, 117, 705, 4972, 39916, 360105, 3606865, 39721266, 477061026, 6205806061, 86925018817, 1304396077272, 20877063837400, 355003736855697, 6391465311099681, 121460116022428510, 2429579599296960430, 51027940329395658981, 1122742916106886416001

Offset: 1

Author

Johannes W. Meijer, Jul 27 2011

Keywords

• nonn
• easy

Comments

a(n)/ceiling(n/2), i.e., a(n) divided by the positive integers repeated, leads to another sequence of integer numbers [1, 1, 3, 11, 39, 235, 1243, 9979, ... ].

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..450

Crossrefs

Cf. A047920, A076732.

Programs

Maple

A193463:=proc(n): add(A076732(n,k), k=1..n) end: A076732:=proc(n,k): (k/(n-k)!)*A047920(n,k) end: A047920:=proc(n,k): add(((-1)^j)*binomial(k-1,j)*(n-1-j)!, j=0..k-1) end: seq(A193463(n), n=1..22);

Mathematica

A000240[n_] := Subfactorial[n] - (-1)^n;
T[n_, k_] := T[n, k] = Switch[k, 1, 1, n, A000240[n], _, k*T[n - 1, k - 1] + T[n - 1, k]];
a[n_] := Sum[T[n, k], {k, 1, n}];
Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Nov 14 2023 *)

Formula

a(n) = Sum_{k=1..n} A076732(n,k).

a(n) = Sum_{k=1..n} (k/(n-k)!)*A047920(n,k).

a(n) = Sum_{k=1..n} (k/(n-k)!) * Sum_{j=0..k-1} (-1)^j*binomial(k-1,j)*(n-1-j)!.