A137339 A triangular sequence from a functional coefficient expansion of a raising factorial type: p(x,t)=1/(1-t)^(m*x);m=3.

1, 0, 3, 0, 3, 9, 0, 6, 27, 27, 0, 18, 99, 162, 81, 0, 72, 450, 945, 810, 243, 0, 360, 2466, 6075, 6885, 3645, 729, 0, 2160, 15876, 43848, 59535, 42525, 15309, 2187, 0, 15120, 117612, 354564, 548289, 476280, 234738, 61236, 6561, 0, 120960, 986256, 3189348

Offset: 1

Author

Roger L. Bagula, Apr 20 2008

Keywords

• nonn
• uned
• tabl

Comments

Row sums are:

{1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800}

Also the Bell transform of A052560. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

Examples

			{1},
{0, 3},
{0, 3, 9},
{0, 6, 27, 27},
{0, 18, 99, 162, 81},
{0, 72, 450, 945, 810, 243},
{0, 360, 2466, 6075, 6885, 3645, 729},
{0, 2160, 15876, 43848, 59535, 42525, 15309, 2187},
{0, 15120, 117612, 354564, 548289, 476280, 234738, 61236, 6561},
{0, 120960, 986256, 3189348, 5450004, 5455107, 3306744, 1194102, 236196, 19683},
{0, 1088640, 9239184, 31662900, 58618080, 65445975, 46126017, 20667150, 5708070, 885735, 59049}
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62 - 63

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> 3*n!, 8); # Peter Luschny, Jan 27 2016

Mathematica

Clear[p, g, m]; m = 3; p[t_] = 1/(1 - t)^(m*x); Table[ ExpandAll[n!SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[3#!&, rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

p(x,t)=1/(1-t)^(m*x)=Sum[s(x,n)*t^n/n!;m=3. out_n,m=n!*Coefficients( s(x,n)).