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A347056
Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
1
1, 3, 4, 6, 16, 10, 10, 40, 50, 20, 15, 80, 150, 120, 35, 21, 140, 350, 420, 245, 56, 28, 224, 700, 1120, 980, 448, 84, 36, 336, 1260, 2520, 2940, 2016, 756, 120, 45, 480, 2100, 5040, 7350, 6720, 3780, 1200, 165, 55, 660, 3300, 9240, 16170, 18480, 13860, 6600, 1815, 220
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
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
Cf. A097805 (p=0), A103406 (p=1), A124932 (essentially p=2).
From Wolfdieter Lang, Sep 30 2021: (Start)
Columns (with leading zeros): A000217(n+1), 4*A000294, 10*A000332(n+2), 20*A000389(n+2), 35*A000579(n+2), 56*A000580(n+2), 84*A000581(n+2), 120*A000582(n+2), ...
Diagonals: A000292(k+1), A004320(k+1), 2*A006411(k+1), 10*A040977, ... (End)
Sequence in context: A369735 A322956 A122727 * A089249 A173944 A092933
KEYWORD
nonn,tabl,easy
AUTHOR
Luc Rousseau, Aug 14 2021
STATUS
approved