A121547 Fourth slice along the 1-2-plane in the cube a(m,n,o) = a(m-1,n,o) + a(m,n-1,o) + a(m,n,o-1) for which the first slice is Pascal's triangle (slice read by antidiagonals).

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, 0, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 0, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 0, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220

Offset: 0

Author

Thomas Wieder, Aug 06 2006

Keywords

• nonn
• tabl

Comments

Essentially the same as triangle A178820 with an additional column of zeros at the left border. - Georg Fischer, Jul 31 2023

Examples

			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
		

Crossrefs

Cf. A000292, A003506, A007318, A094305, A121306, A119800, A178820.

Programs

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

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.

Extensions

a(55)-a(56) corrected and more terms from Georg Fischer, Jul 31 2023