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%I #14 Feb 26 2019 19:15:49
%S 1,0,2,0,2,4,0,4,12,8,0,12,44,48,16,0,48,200,280,160,32,0,240,1096,
%T 1800,1360,480,64,0,1440,7056,12992,11760,5600,1344,128,0,10080,52272,
%U 105056,108304,62720,20608,3584,256,0,80640,438336,944992,1076544,718368,290304,69888,9216,512
%N Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.
%C Row sums are factorials.
%C Also the Bell transform of A052849 (with a(0)=2). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62-63
%F From _Peter Luschny_, Feb 26 2019: (Start)
%F p(n, x) = n!*Sum_{k=0..n} (-1)^n*binomial(-x, k)*binomial(-x, n-k).
%F p(n, x) = (n + 2*x - 1)!/(2*x - 1)!.
%F T(n, k) = [x^k] p(n,x). (End)
%e [0] {1},
%e [1] {0, 2},
%e [2] {0, 2, 4},
%e [3] {0, 4, 12, 8},
%e [4] {0, 12, 44, 48, 16},
%e [5] {0, 48, 200, 280, 160, 32},
%e [6] {0, 240, 1096, 1800, 1360, 480, 64},
%e [7] {0, 1440, 7056, 12992, 11760, 5600, 1344, 128},
%e [8] {0, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256},
%e [9] {0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512}.
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> `if`(n<2,2,2*n!), 8); # _Peter Luschny_, Jan 27 2016
%p p := (n,x) -> (n + 2*x - 1)!/(2*x - 1)!:
%p seq(seq(coeff(expand(p(n,x)), x, k), k=0..n), n=0..9); # _Peter Luschny_, Feb 26 2019
%t m = 2; p[x, 0] = 1; p[x, -1] = 0; p[x, 1] = m*x;
%t p[x_, n_] := p[x, n] = (m*x + n - 1)*p[x, n - 1];
%t Table[CoefficientList[p[x, n], x], {n, 0, 9}] // Flatten
%t (* Second program: *)
%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t B = BellMatrix[Function[n, If[n < 2, 2, 2*n!]], rows = 12];
%t Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%Y Apart from signs, same as A137312.
%K nonn,tabl
%O 0,3
%A _Roger L. Bagula_, Apr 20 2008
%E Edited and offset set to 0 by _Peter Luschny_, Feb 26 2019
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