OFFSET
0,2
COMMENTS
This triangle is T[3] in the sequence (T[p]) of triangles defined by: T[p](n,k) = (k+p)*(n+p-1)!/(k!*(n-k)!*p!) and T[0](0,0)=1.
Riordan triangle (1/(1-x)^3, x/(1-x)) with column k scaled with A000292(k+1) = binomial(k+3, 3), for k >= 0. - Wolfdieter Lang, Sep 30 2021
LINKS
FORMULA
T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6.
G.f. column k: x^k*binomial(k+3, 3)/(1 - x)^(k+3), for k >= 0. - Wolfdieter Lang, Sep 30 2021
EXAMPLE
T(6,2) = (6+1)*(6+2)*(2+3)*binomial(6,2)/6 = 7*8*5*15/6 = 700.
The triangle T begins:
n \ k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 3 4
2: 6 16 10
3: 10 40 50 20
4: 15 80 150 120 35
5: 21 140 350 420 245 56
6: 28 224 700 1120 980 448 84
7: 36 336 1260 2520 2940 2016 756 120
8: 45 480 2100 5040 7350 6720 3780 1200 165
9: 55 660 3300 9240 16170 18480 13860 6600 1815 220
10: 66 880 4950 15840 32340 44352 41580 26400 10890 2640 286
... - Wolfdieter Lang, Sep 30 2021
PROG
(PARI)
T(p, n, k)=if(n==0&&p==0, 1, ((k+p)*(n+p-1)!)/(k!*(n-k)!*p!))
for(n=0, 9, for(k=0, n, print1(T(3, n, k), ", ")))
CROSSREFS
KEYWORD
AUTHOR
Luc Rousseau, Aug 14 2021
STATUS
approved