A249163 Triangle read by rows: the positive terms of A163626.

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Author

Paul Curtz, Dec 15 2014

Keywords

• nonn
• tabf

Comments

We have two possibilities: with or without 0's.

Without 0's:

1,

1,

1, 2,

1, 12,

1, 50, 24,

1, 180, 360,

etc.

Sum of every row: A000670(n).

First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.

With 0's:

1, 0, 0, 0,

1, 0, 0, 0,

1, 2, 0, 0,

1, 12, 0, 0,

1, 50, 24, 0,

1, 180, 360, 0,

1, 602, 3360, 720,

etc.

The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Crossrefs

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Programs

Mathematica

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)