A178822 -id:A178822 - cp's OEIS frontend

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1

Offset: 0

Harlan J. Brothers, Jun 16 2010

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P for n = {1, 2, 3, ..., k}.

			Prism begins (levels 1-4):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Level 1 = A007318.
Level 2 = A003506.
Level 3 = A094305.
Level 4 = A178820.
Level 5 = A178821.
Level 6 = A178822.
Sums of shallow diagonals for each level correspond to rows of square A037027.
Contains A109649 and A046816.
P = A000984.
P = A006480.
P = A000897.
P<3n-2, 3n-2, n> = A113424.

end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).

Keyword tabf by Michel Marcus, Oct 22 2017

A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 5, 5, 15, 30, 15, 35, 105, 105, 35, 70, 280, 420, 280, 70, 126, 630, 1260, 1260, 630, 126, 210, 1260, 3150, 4200, 3150, 1260, 210, 330, 2310, 6930, 11550, 11550, 6930, 2310, 330, 495, 3960, 13860, 27720, 34650, 27720, 13860, 3960, 495, 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715

Offset: 0

Harlan J. Brothers, Jun 19 2010

The product of the pentatope numbers (A000332, beginning with fifth term) and Pascal's triangle (A007318). Also level 5 of Pascal's prism (A178819) read by rows: (i+4; 4, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
   1;
   5,   5;
  15,  30,  15;
  35, 105, 105,  35;
  70, 280, 420, 280,  70;

G. C. Greubel, Rows n=0..100 of triangle, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000332, A007318, A178819, A178820, A178822.

Rows sum to A003472, shallow diagonals sum to A001873.

T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+4, 4)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

/* As triangle */ [[Binomial(n+4,4)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017

T:=(n,k)->binomial(n+4,4)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019

Table[Multinomial[4, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

{T(n,k) = binomial(n+4, 4)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019

[[binomial(n+4, 4)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

T(n,k) = C(n+4,4) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(5, i, j) = (i+3; 4, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^5. - Ilya Gutkovskiy, Mar 20 2020

cp's OEIS Frontend

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions