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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. 1

%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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Last modified March 29 03:41 EDT 2024. Contains 371264 sequences. (Running on oeis4.)