%I #14 Mar 14 2020 11:26:18
%S 1,-1,1,19,-39,20,-1513,4705,-4872,1680,315523,-1314807,2052644,
%T -1422960,369600,-136085041,710968441,-1484552160,1548707160,
%U -807206400,168168000,105261234643,-661231439271,1729495989332,-2410936679424,1889230062720,-789044256000,137225088000
%N Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.
%C The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See A318259 and the cross-references for some other members.
%F Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1
%F and S(n,k) = [x^k]P(3,n), then T(n,k) = Sum_{j=0..k}((-1)^(k-j)*binomial(n-j, n-k)* Sum_{i=0..n}((-1)^i*S(n,i)*binomial(n-i,j))).
%e [0] [ 1]
%e [1] [ -1, 1]
%e [2] [ 19, -39, 20]
%e [3] [ -1513, 4705, -4872, 1680]
%e [4] [ 315523, -1314807, 2052644, -1422960, 369600]
%e [5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]
%o (Sage) # uses[EW from A318259]
%o def A318260row(n): return EW(3, n)
%o print(flatten([A318260row(n) for n in (0..6)]))
%Y Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.
%Y Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).
%K sign,tabl
%O 0,4
%A _Peter Luschny_, Sep 06 2018
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