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

%I A137339 #7 Jun 28 2018 08:24:56
%S A137339 1,0,3,0,3,9,0,6,27,27,0,18,99,162,81,0,72,450,945,810,243,0,360,2466,
%T A137339 6075,6885,3645,729,0,2160,15876,43848,59535,42525,15309,2187,0,15120,
%U A137339 117612,354564,548289,476280,234738,61236,6561,0,120960,986256,3189348
%N A137339 A triangular sequence from a functional coefficient expansion of a raising factorial type: p(x,t)=1/(1-t)^(m*x);m=3.
%C A137339 Row sums are:
%C A137339 {1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800}
%C A137339 Also the Bell transform of A052560. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D A137339 Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62 - 63
%F A137339 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)).
%e A137339 {1},
%e A137339 {0, 3},
%e A137339 {0, 3, 9},
%e A137339 {0, 6, 27, 27},
%e A137339 {0, 18, 99, 162, 81},
%e A137339 {0, 72, 450, 945, 810, 243},
%e A137339 {0, 360, 2466, 6075, 6885, 3645, 729},
%e A137339 {0, 2160, 15876, 43848, 59535, 42525, 15309, 2187},
%e A137339 {0, 15120, 117612, 354564, 548289, 476280, 234738, 61236, 6561},
%e A137339 {0, 120960, 986256, 3189348, 5450004, 5455107, 3306744, 1194102, 236196, 19683},
%e A137339 {0, 1088640, 9239184, 31662900, 58618080, 65445975, 46126017, 20667150, 5708070, 885735, 59049}
%p A137339 # The function BellMatrix is defined in A264428.
%p A137339 BellMatrix(n -> 3*n!, 8); # _Peter Luschny_, Jan 27 2016
%t A137339 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]
%t A137339 (* Second program: *)
%t A137339 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A137339 B = BellMatrix[3#!&, rows = 12];
%t A137339 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%K A137339 nonn,uned,tabl
%O A137339 1,3
%A A137339 _Roger L. Bagula_, Apr 20 2008