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

0,3

Comments

Row sums are factorials.

Also the Bell transform of A052849 (with a(0)=2). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

References

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 62-63

Links

Table of n, a(n) for n=0..54.

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)

Example

[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}.

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 *)

Crossrefs

Apart from signs, same as A137312.

Sequence in context: A126440 A131186 A137312 * A263399 A143507 A172040

Adjacent sequences: A137317 A137318 A137319 * A137321 A137322 A137323

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 20 2008

Extensions

Edited and offset set to 0 by Peter Luschny, Feb 26 2019

Status

approved