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

1, -1, 1, 19, -39, 20, -1513, 4705, -4872, 1680, 315523, -1314807, 2052644, -1422960, 369600, -136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000, 105261234643, -661231439271, 1729495989332, -2410936679424, 1889230062720, -789044256000, 137225088000

Offset: 0

Author

Peter Luschny, Sep 06 2018

Keywords

• sign
• tabl

Comments

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.

Examples

			[0] [         1]
[1] [        -1,         1]
[2] [        19,       -39,          20]
[3] [     -1513,      4705,       -4872,       1680]
[4] [    315523,  -1314807,     2052644,   -1422960,     369600]
[5] [-136085041, 710968441, -1484552160, 1548707160, -807206400, 168168000]
		

Crossrefs

Cf. T(n,0) ~ A002115(n) (signed), T(n,n) = A014606.

Cf. A167374 (m=0), A028246 & A163626 (m=1), A318259 (m=2), this seq (m=3).

Programs

Sage

# uses[EW from A318259]
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))

Formula

Let P(m,n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x with P(m,0) = 1

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))).