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, 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}.
.
Row sums start: 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...
References
- Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 56-57.
Crossrefs
Apart from signs, same as A137320.
Programs
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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
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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-François Alcover, Jun 28 2018, after Peter Luschny *)
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)
Extensions
Edited and offset set to 0 by Peter Luschny, Feb 26 2019
Comments