|
|
A318260
|
|
Generalized Worpitzky numbers W_{m}(n,k) for m = 3, n >= 0 and 0 <= k <= n, triangle read by rows.
|
|
1
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
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.
|
|
LINKS
|
|
|
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))).
|
|
EXAMPLE
|
[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]
|
|
PROG
|
def A318260row(n): return EW(3, n)
print(flatten([A318260row(n) for n in (0..6)]))
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|