A054649 Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.
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
Examples
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)
Links
- Seiichi Manyama, Rows n = 0..139, flattened
Programs
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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
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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 *) -
PARI
row(n) = Vec(n!*sum(k=0, n, binomial(x-n, k))); \\ Seiichi Manyama, Sep 24 2021
Formula
T(n, k) = n! * [x^(n - k)] hypergeom([-n, -x + n], [-n], -1). - Peter Luschny, Nov 27 2021