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A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.

%I A318260 #14 Mar 14 2020 11:26:18
%S A318260 1,-1,1,19,-39,20,-1513,4705,-4872,1680,315523,-1314807,2052644,
%T A318260 -1422960,369600,-136085041,710968441,-1484552160,1548707160,
%U A318260 -807206400,168168000,105261234643,-661231439271,1729495989332,-2410936679424,1889230062720,-789044256000,137225088000
%N A318260 Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.
%C A318260 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 A318260 Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1
%F A318260 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 A318260 [0] [         1]
%e A318260 [1] [        -1,         1]
%e A318260 [2] [        19,       -39,          20]
%e A318260 [3] [     -1513,      4705,       -4872,       1680]
%e A318260 [4] [    315523,  -1314807,     2052644,   -1422960,     369600]
%e A318260 [5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]
%o A318260 (Sage) # uses[EW from A318259]
%o A318260 def A318260row(n): return EW(3, n)
%o A318260 print(flatten([A318260row(n) for n in (0..6)]))
%Y A318260 Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.
%Y A318260 Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).
%K A318260 sign,tabl
%O A318260 0,4
%A A318260 _Peter Luschny_, Sep 06 2018