A137320 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.
1, 0, 2, 0, 2, 4, 0, 4, 12, 8, 0, 12, 44, 48, 16, 0, 48, 200, 280, 160, 32, 0, 240, 1096, 1800, 1360, 480, 64, 0, 1440, 7056, 12992, 11760, 5600, 1344, 128, 0, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256, 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512
Offset: 0
Examples
[0] {1},
[1] {0, 2},
[2] {0, 2, 4},
[3] {0, 4, 12, 8},
[4] {0, 12, 44, 48, 16},
[5] {0, 48, 200, 280, 160, 32},
[6] {0, 240, 1096, 1800, 1360, 480, 64},
[7] {0, 1440, 7056, 12992, 11760, 5600, 1344, 128},
[8] {0, 10080, 52272, 105056, 108304, 62720, 20608, 3584, 256},
[9] {0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512}.
References
- Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62-63
Crossrefs
Apart from signs, same as A137312.
Programs
-
Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> `if`(n<2,2,2*n!), 8); # Peter Luschny, Jan 27 2016 p := (n,x) -> (n + 2*x - 1)!/(2*x - 1)!: seq(seq(coeff(expand(p(n,x)), x, k), k=0..n), n=0..9); # Peter Luschny, Feb 26 2019
-
Mathematica
m = 2; p[x, 0] = 1; p[x, -1] = 0; p[x, 1] = m*x; p[x_, n_] := p[x, n] = (m*x + n - 1)*p[x, n - 1]; Table[CoefficientList[p[x, n], x], {n, 0, 9}] // Flatten (* 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[Function[n, If[n < 2, 2, 2*n!]], rows = 12]; Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
Formula
From Peter Luschny, Feb 26 2019: (Start)
p(n, x) = n!*Sum_{k=0..n} (-1)^n*binomial(-x, k)*binomial(-x, n-k).
p(n, x) = (n + 2*x - 1)!/(2*x - 1)!.
T(n, k) = [x^k] p(n,x). (End)
Extensions
Edited and offset set to 0 by Peter Luschny, Feb 26 2019
Comments