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A054649
Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.
5
1, 1, 0, 1, -3, 4, 1, -9, 32, -36, 1, -18, 131, -426, 528, 1, -30, 375, -2370, 7544, -9600, 1, -45, 865, -8955, 52414, -163800, 213120, 1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240, 1, -84, 3122, -66696, 893249, -7664916, 41096908, -125747664, 167973120
OFFSET
0,5
LINKS
FORMULA
T(n, k) = n! * [x^(n - k)] hypergeom([-n, -x + n], [-n], -1). - Peter Luschny, Nov 27 2021
EXAMPLE
Triangle begins:
1;
1, 0;
1, -3, 4;
1, -9, 32, -36;
1, -18, 131, -426, 528;
1, -30, 375, -2370, 7544, -9600;
1, -45, 865, -8955, 52414, -163800, 213120;
1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240;
...
From Peter Luschny, Nov 27 2021: (Start)
The row reversed triangle can be seen as the coefficients of a sequence of monic polynomials with monomials sorted in ascending order which start:
[0] 1;
[1] x;
[2] 4 - 3*x + x^2;
[3] -36 + 32*x - 9*x^2 + x^3;
[4] 528 - 426*x + 131*x^2 - 18*x^3 + x^4;
[5] -9600 + 7544*x - 2370*x^2 + 375*x^3 - 30*x^4 + x^5; (End)
MAPLE
# Some older Maple versions are known to have a bug in the hypergeom function.
with(ListTools): with(PolynomialTools):
CoeffList := p -> op(Reverse(CoefficientList(simplify(p), x))):
p := k -> k!*hypergeom([-k, -x + k], [-k], -1):
seq(CoeffList(p(k)), k = 0..8); # Peter Luschny, Nov 27 2021
MATHEMATICA
c[n_, k_] := Product[n-i, {i, 0, k-1}]/k!; row[n_] := CoefficientList[ n!*Sum[c[x-n, k], {k, 0, n}], x] // Reverse; Table[ row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Oct 04 2012 *)
PROG
(PARI) row(n) = Vec(n!*sum(k=0, n, binomial(x-n, k))); \\ Seiichi Manyama, Sep 24 2021
CROSSREFS
KEYWORD
sign,tabl,nice
AUTHOR
N. J. A. Sloane, Apr 16 2000
STATUS
approved