A178820 -id:A178820 - 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

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 6, 6, 21, 42, 21, 56, 168, 168, 56, 126, 504, 756, 504, 126, 252, 1260, 2520, 2520, 1260, 252, 462, 2772, 6930, 9240, 6930, 2772, 462, 792, 5544, 16632, 27720, 27720, 16632, 5544, 792, 1287, 10296, 36036, 72072, 90090, 72072, 36036, 10296, 1287

Offset: 0

Harlan J. Brothers, Jun 19 2010

The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
    1;
    6,   6;
   21,  42,  21;
   56, 168, 168,  56;
  126, 504, 756, 504, 126;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000389, A007318, A178819, A178820, A178821.

Rows sum to A054849, shallow diagonals sum to A001874.

/* As triangle */ [[Binomial(n+5,5)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Oct 23 2017

Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
Table[Binomial[n + 5, 5]*Binomial[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

for(n=0,10, for(k=0,n, print1(binomial(n+5,5)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 25 2017

T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^6. - Ilya Gutkovskiy, Mar 20 2020

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).

Original entry on oeis.org

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

Thomas Wieder, Aug 06 2006

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

			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

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

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

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.

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

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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.

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

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A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

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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).

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