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Triangle of coefficients of Bateman polynomial n!Z_n(-x).
1

%I #12 Jun 04 2019 02:58:31

%S 1,1,2,2,12,6,6,72,90,20,24,480,1080,560,70,120,3600,12600,11200,3150,

%T 252,720,30240,151200,201600,94500,16632,924,5040,282240,1905120,

%U 3528000,2425500,698544,84084,3432,40320,2903040,25401600,62092800,58212000,24216192,4708704,411840,12870

%N Triangle of coefficients of Bateman polynomial n!Z_n(-x).

%e {1};

%e {1, 2};

%e {2, 12, 6};

%e {6, 72, 90, 20};

%e {24, 480, 1080, 560, 70}; ...

%e 2!Z_2(-x) = 2+12x+6x^2.

%p A073768_row := proc(n) n!*hypergeom([-n, n+1], [1, 1], -x);

%p PolynomialTools:-CoefficientList(simplify(%), x) end:

%p seq(A073768_row(n), n=0..8): ListTools[FlattenOnce]([%]); # _Peter Luschny_, Jan 31 2016

%t T[n_, k_] := n! (n+k)! / (n-k)! / k!^3;

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 04 2019 *)

%o (PARI) {T(n, k) = if( k<0 || k>n, 0, n! * (n+k)! / (n-k)! / k!^3)}

%Y A073767 gives row sums.

%K nonn,tabl

%O 0,3

%A _Michael Somos_, Aug 08 2002