OFFSET
0,5
COMMENTS
Essentially the same as triangle A178820 with an additional column of zeros at the left border. - Georg Fischer, Jul 31 2023
FORMULA
a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) with initialization values a(1,0,0) = 1 and a(m<>1=0, n>=0, 0>=o) = 0.
EXAMPLE
The second row is 1, 4, 10, 20, 35, 56, 84, 120, 165, 220 = A000292, i.e., Tetrahedral (or pyramidal) numbers: binomial(n+2,3) = n(n+1)(n+2)/6 (core).
The third row is 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860 = A033488 = n*(n+1)*(n+2)*(n+3)/6.
The main diagonal is 0, 4, 60, 560, 4200, 27720, 168168, 960960, 5250960, 27713400 = {0} U A002803*4.
Triangle starts:
0
0, 1
0, 4, 4
0, 10, 20, 10
0, 20, 60, 60, 20
0, 35, 140, 210, 140, 35
0, 56, 280, 560, 560, 280, 56
0, 84, 504, 1260, 1680, 1260, 504, 84
MAPLE
T:=(n, k)->binomial(n+3, 3)*binomial(n, k): seq(print(seq(T(n-1, k-1), k=0..n)), n=0..10); # Georg Fischer, Jul 31 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Wieder, Aug 06 2006
EXTENSIONS
a(55)-a(56) corrected and more terms from Georg Fischer, Jul 31 2023
STATUS
approved