A137320 - OEIS

%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