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%I #14 Feb 26 2019 19:15:40
%S 1,0,2,0,-2,4,0,4,-12,8,0,-12,44,-48,16,0,48,-200,280,-160,32,0,-240,
%T 1096,-1800,1360,-480,64,0,1440,-7056,12992,-11760,5600,-1344,128,0,
%U -10080,52272,-105056,108304,-62720,20608,-3584,256,0,80640,-438336,944992,-1076544,718368,-290304,69888,-9216,512
%N Coefficients of generalized factorial polynomials p(x, n) = (x/a - (n-1))*p(x, n-1) with p(x, 0) = 1, p(x, 1) = x/a and a = 1/2. Triangle read by rows, for n >= 0 and 0 <= k <= n.
%C The polynomials are defined by a recurrence given by S. Roman (see reference).
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 56-57.
%F From _Peter Luschny_, Feb 26 2019: (Start)
%F p(x, n) = n!*Sum_{k=0..n} binomial(x, k)*binomial(x, n-k).
%F p(x, n) = (-1)^n*(n - 2*x - 1)!/(-2*x - 1)!.
%F T(n, k) = [x^k] p(x, n). (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}.
%e .
%e Row sums start: 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...
%p BellMatrix(n -> `if`(n<2,(-1)^n*2,(-1)^n*2*n!), 8); # _Peter Luschny_, Jan 27 2016
%p p := (n, x) -> ((-1)^n*(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 a = 1/2; p[x, 0] = 1; p[x, 1] = x/a;
%t p[x_, n_] := p[x, n] = (x/a - (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, (-1)^n*2, (-1)^n*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 A137320.
%K tabl,sign
%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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