Edited by _N. J. A. Sloane, _, Jun 02 2009

_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Dec 30 2008

Triangular row sum sequence recursion Eulerian number level scale 5Triangle read by rows: A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (n + 1)*(n + 2)*A(n - 2, k - 1).

Row sums are (2*(n + 3)!) except for n=1 which is 2: {2, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600}.

{2, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200,

12454041600}.

This sequence is designed to be at the Eulerian numbers equivalent level

for a scale five row sum type.

A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (n + 1)*(n + 2)*A(n - 2, k - 1).

nonn,uned,tabl,new

Edited by N. J. A. Sloane, Jun 02 2009

Triangular row sum sequence recursion Eulerian number level scale 5: A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (n + 1)*(n + 2)*A(n - 2, k - 1).

2, 120, 120, 2, 1436, 2, 2, 5038, 5038, 2, 2, 5124, 70388, 5124, 2, 2, 5238, 357640, 357640, 5238, 2, 2, 5384, 731806, 5783216, 731806, 5384, 2, 2, 5566, 1208610, 38702622, 38702622, 1208610, 5566, 2, 2, 5788, 1806416, 120409892, 713559004

1,1

Row sums are (2*(n + 3)!) except for n=1 which is 2:

{2, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200,

12454041600}.

This sequence is designed to be at the Eulerian numbers equivalent level

for a scale five row sum type.

A(n,k)=A(n - 1, k - 1) + A(n - 1, k) + (n + 1)*(n + 2)*A(n - 2, k - 1).

{2},

{120, 120},

{2, 1436, 2},

{2, 5038, 5038, 2},

{2, 5124, 70388, 5124, 2},

{2, 5238, 357640, 357640, 5238, 2},

{2, 5384, 731806, 5783216, 731806, 5384, 2},

{2, 5566, 1208610, 38702622, 38702622, 1208610, 5566, 2},

{2, 5788, 1806416, 120409892, 713559004, 120409892, 1806416, 5788, 2},

{2, 6054, 2546916, 281752828, 5942715000, 5942715000, 281752828, 2546916, 6054, 2}

Clear[A]; A[2, 1] := A[2, 2] = (5)!;

A[3, 2] = 2*(6)! - 4; A[4, 2] = A[4, 3] = (7)! - 2;

A[n_, 1] := 2; A[n_, n_] := 2;

A[n_, k_] := A[n - 1, k - 1] + A[n - 1, k] + (n + 1)*(n + 2)*A[n - 2, k - 1];

a = Table[A[n, k], {n, 10}, {k, n}]; Flatten[a]

Table[Apply[Plus, a[[n]]], {n, 1, 10}];

Table[Apply[Plus, a[[n]]]/(2*(n + 3)!), {n, 1, 10}]

nonn,uned,tabl,new

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 30 2008

approved