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

0,5

Comments

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

Links

Table of n, a(n) for n=0..65.

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

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

Sequence in context: A236922 A021698 A199739 * A028626 A205507 A137862

Adjacent sequences: A121544 A121545 A121546 * A121548 A121549 A121550

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