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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.

%I A054649 #29 Nov 27 2021 09:19:40
%S A054649 1,1,0,1,-3,4,1,-9,32,-36,1,-18,131,-426,528,1,-30,375,-2370,7544,
%T A054649 -9600,1,-45,865,-8955,52414,-163800,213120,1,-63,1729,-26565,245854,
%U A054649 -1366932,4220376,-5574240,1,-84,3122,-66696,893249,-7664916,41096908,-125747664,167973120
%N A054649 Triangle T(n, k) giving coefficients in expansion of n! * Sum_{i=0..n} binomial(x - n, i) in powers of x.
%H A054649 Seiichi Manyama, <a href="/A054649/b054649.txt">Rows n = 0..139, flattened</a>
%F A054649 T(n, k) = n! * [x^(n - k)] hypergeom([-n, -x + n], [-n], -1). - _Peter Luschny_, Nov 27 2021
%e A054649 Triangle begins:
%e A054649   1;
%e A054649   1,   0;
%e A054649   1,  -3,    4;
%e A054649   1,  -9,   32,    -36;
%e A054649   1, -18,  131,   -426,    528;
%e A054649   1, -30,  375,  -2370,   7544,    -9600;
%e A054649   1, -45,  865,  -8955,  52414,  -163800,  213120;
%e A054649   1, -63, 1729, -26565, 245854, -1366932, 4220376, -5574240;
%e A054649   ...
%e A054649 From _Peter Luschny_, Nov 27 2021: (Start)
%e A054649 The row reversed triangle can be seen as the coefficients of a sequence of monic polynomials with monomials sorted in ascending order which start:
%e A054649 [0]     1;
%e A054649 [1]              x;
%e A054649 [2]     4 -    3*x +      x^2;
%e A054649 [3]   -36 +   32*x -    9*x^2 +     x^3;
%e A054649 [4]   528 -  426*x +  131*x^2 -  18*x^3 +    x^4;
%e A054649 [5] -9600 + 7544*x - 2370*x^2 + 375*x^3 - 30*x^4 + x^5; (End)
%p A054649 # Some older Maple versions are known to have a bug in the hypergeom function.
%p A054649 with(ListTools): with(PolynomialTools):
%p A054649 CoeffList := p -> op(Reverse(CoefficientList(simplify(p), x))):
%p A054649 p := k -> k!*hypergeom([-k, -x + k], [-k], -1):
%p A054649 seq(CoeffList(p(k)), k = 0..8); # _Peter Luschny_, Nov 27 2021
%t A054649 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 *)
%o A054649 (PARI) row(n) = Vec(n!*sum(k=0, n, binomial(x-n, k))); \\ _Seiichi Manyama_, Sep 24 2021
%Y A054649 Cf. A008275, A008276, A048994, A054651, A054655.
%K A054649 sign,tabl,nice
%O A054649 0,5
%A A054649 _N. J. A. Sloane_, Apr 16 2000