%I #34 Sep 24 2021 13:42:46
%S 1,1,-1,1,-5,6,1,-12,47,-60,1,-22,179,-638,840,1,-35,485,-3325,11274,
%T -15120,1,-51,1075,-11985,74524,-245004,332640,1,-70,2086,-34300,
%U 336049,-1961470,6314664,-8648640,1,-92,3682,-83720,1182769
%N Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.
%H Robert Israel, <a href="/A054655/b054655.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened).
%F n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
%F From _Robert Israel_, Jul 12 2016: (Start)
%F G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).
%F E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)
%e Triangle begins:
%e 1;
%e 1, -1;
%e 1, -5, 6;
%e 1, -12, 47, -60;
%e 1, -22, 179, -638, 840;
%e 1, -35, 485, -3325, 11274, -15120;
%e 1, -51, 1075, -11985, 74524, -245004, 332640;
%e 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
%e ...
%p a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x,n)),x,n-k),k=0..n) end: # _Peter Luschny_, Nov 28 2010
%t row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* _Jean-François Alcover_, Apr 06 2012, after Maple *)
%o (PARI) T(n,k)=polcoef(n!*binomial(x-n,n), n-k);
%Y Cf. A000407, A054649, A054651, A054654, A008276.
%K sign,tabl,easy,nice
%O 0,5
%A _N. J. A. Sloane_, Apr 18 2000