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A137312 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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The polynomials are defined by a recurrence given by S. Roman (see reference).

REFERENCES

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 56-57.

LINKS

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

FORMULA

From Peter Luschny, Feb 26 2019: (Start)

p(x, n) = n!*Sum_{k=0..n} binomial(x, k)*binomial(x, n-k).

p(x, n) = (-1)^n*(n - 2*x - 1)!/(-2*x - 1)!.

T(n, k) = [x^k] p(x, n). (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}.

.

Row sums start: 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...

MAPLE

BellMatrix(n -> `if`(n<2, (-1)^n*2, (-1)^n*2*n!), 8); # Peter Luschny, Jan 27 2016

p := (n, x) -> ((-1)^n*(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

a = 1/2; p[x, 0] = 1; p[x, 1] = x/a;

p[x_, n_] := p[x, n] = (x/a - (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, (-1)^n*2, (-1)^n*2*n!]], rows = 12];

Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 28 2018, after Peter Luschny *)

CROSSREFS

Apart from signs, same as A137320.

Sequence in context: A209697 A126440 A131186 * A137320 A263399 A143507

Adjacent sequences:  A137309 A137310 A137311 * A137313 A137314 A137315

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Apr 20 2008

EXTENSIONS

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

STATUS

approved

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Last modified May 14 05:45 EDT 2021. Contains 343872 sequences. (Running on oeis4.)