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

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

0,2

Comments

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.

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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

Prog

(Magma) /* As triangle */ [[Binomial(n+4, 4)*Binomial(n, k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017
(PARI) {T(n, k) = binomial(n+4, 4)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019
(Sage) [[binomial(n+4, 4)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019
(GAP) T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+4, 4)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

Crossrefs

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

Rows sum to A003472, shallow diagonals sum to A001873.

Sequence in context: A050341 A140360 A302511 * A145599 A275016 A189316

Adjacent sequences: A178818 A178819 A178820 * A178822 A178823 A178824

Keyword

easy,nonn,tabl

Author

Harlan J. Brothers, Jun 19 2010

Status

approved