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A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

%I A054655 #37 May 23 2025 01:14:02
%S A054655 1,1,-1,1,-5,6,1,-12,47,-60,1,-22,179,-638,840,1,-35,485,-3325,11274,
%T A054655 -15120,1,-51,1075,-11985,74524,-245004,332640,1,-70,2086,-34300,
%U A054655 336049,-1961470,6314664,-8648640,1,-92,3682,-83720,1182769
%N A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.
%H A054655 Robert Israel, <a href="/A054655/b054655.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened).
%H A054655 Takao Komatsu, <a href="https://arxiv.org/abs/2505.09357">Some explicit values of a q-multiple zeta function at roots of unity</a>, arXiv:2505.09357 [math.NT], 2025. See p. 10.
%F A054655 n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
%F A054655 From _Robert Israel_, Jul 12 2016: (Start)
%F A054655 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 A054655 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 A054655 Triangle begins:
%e A054655   1;
%e A054655   1,  -1;
%e A054655   1,  -5,    6;
%e A054655   1, -12,   47,    -60;
%e A054655   1, -22,  179,   -638,    840;
%e A054655   1, -35,  485,  -3325,  11274,   -15120;
%e A054655   1, -51, 1075, -11985,  74524,  -245004,  332640;
%e A054655   1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
%e A054655   ...
%p A054655 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 A054655 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 A054655 (PARI) T(n,k)=polcoef(n!*binomial(x-n,n), n-k);
%Y A054655 Cf. A000407, A054649, A054651, A054654, A008276.
%K A054655 sign,tabl,easy,nice
%O A054655 0,5
%A A054655 _N. J. A. Sloane_, Apr 18 2000