A343960 Triangle read by rows: T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m), n >= m >= 1.

1, 1, 2, 1, 5, 4, 1, 9, 17, 8, 1, 14, 46, 49, 16, 1, 20, 100, 180, 129, 32, 1, 27, 190, 510, 603, 321, 64, 1, 35, 329, 1225, 2121, 1827, 769, 128, 1, 44, 532, 2618, 6202, 7700, 5164, 1793, 256, 1, 54, 816, 5124, 15876, 26628, 25392, 13878, 4097, 512

Offset: 1

Author

Yuriy Shablya, May 05 2021

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  ---------------------------------------------------------------------
   n \ m |     1     2     3     4     5     6     7     8     9    10
  -------+-------------------------------------------------------------
   1     |     1
   2     |     1     2
   3     |     1     5     4
   4     |     1     9    17     8
   5     |     1    14    46    49    16
   6     |     1    20   100   180   129    32
   7     |     1    27   190   510   603   321    64
   8     |     1    35   329  1225  2121  1827   769   128
   9     |     1    44   532  2618  6202  7700  5164  1793   256
   10    |     1    54   816  5124 15876 26628 25392 13878  4097   512
		

Crossrefs

Cf. A001263.

Programs

Mathematica

T[n_, m_] := Sum[Binomial[n, m - k] * Binomial[n, m] * k/n, {k, 1, n}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, May 06 2021 *)

Maxima

T(n,m):=sum((k/n)*binomial(n,m-k)*binomial(n,m),k,1,m)

Formula

T(n,m) = Sum_{k=1..m} (k/n)*binomial(n,m-k)*binomial(n,m).

G.f.: N(x,y)/(1-N(x,y)), where N(x,y) is a g.f. for the Narayana numbers A001263.

T(n, m) = A001263(n, m)*hypergeom([1 - m, 2], [n - m + 2], -1). - Peter Luschny, May 06 2021