A174158 Triangle read by rows: T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.

1, 1, 1, 1, 9, 1, 1, 36, 36, 1, 1, 100, 400, 100, 1, 1, 225, 2500, 2500, 225, 1, 1, 441, 11025, 30625, 11025, 441, 1, 1, 784, 38416, 240100, 240100, 38416, 784, 1, 1, 1296, 112896, 1382976, 3111696, 1382976, 112896, 1296, 1, 1, 2025, 291600, 6350400, 28005264, 28005264, 6350400, 291600, 2025, 1

Offset

1,5

Links

Stefano Spezia, First 150 rows of the triangle, flattened

Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Enumeration of parallelogram polycubes, arXiv:2105.00971 [cs.DM], 2021.

Formula

T(n,m) = (binomial(n - 1, m - 1)*binomial(n, m - 1)/m)^2.

T(n,m) = A001263(n,m)^2.

T(n,m) = A000290(A007318(n - 1, m - 1)*A007318(n, m - 1)/m). - Stefano Spezia, Dec 23 2018

Example

n\m | 1         2         3         4         5         6         7
----|--------------------------------------------------------------
  1 | 1
  2 | 1         1
  3 | 1         9         1
  4 | 1        36        36         1
  5 | 1       100       400       100         1
  6 | 1       225      2500      2500       225         1
  7 | 1       441     11025     30625     11025       441         1

Maple

a := (n, m) -> binomial(n-1, m-1)^2*binomial(n, m-1)^2/m^2: seq(seq(a(n, m), m = 1 .. n), n = 1 .. 10) # Stefano Spezia, Dec 23 2018

Mathematica

T[n_, m_] = (Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m)^2; Flatten[Table[Table[t[n, m], {m, 1, n}], {n, 1, 10}]]

Prog

(GAP) Flat(List([1..10], n->List([1..n], m->(Binomial(n-1, m-1)*Binomial(n, m-1)/m)^2))); # Stefano Spezia, Dec 23 2018
(PARI)
T(n, m)= (binomial(n-1, m-1)*binomial(n, m-1)/m)^2;
tabl(nn) = for(n=1, nn, for(m=1, n, print1(T(n, m), ", ")); print);
tabl(10) \\ Stefano Spezia, Dec 23 2018

Crossrefs

Cf. A001263 (Narayana numbers), A007318.

Cf. A319743 (row sums).

Sequence in context: A171822 A176490 A370232 * A181144 A142468 A359313

Adjacent sequences: A174155 A174156 A174157 * A174159 A174160 A174161

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 10 2010

Extensions

Edited by Stefano Spezia, Dec 23 2018

Status

approved