A174102 Triangle read by rows: T(n, m) = floor(binomial(n+1, m)* binomial(n+2, m)/(2*m+2)), 1 <= m <= n.

1, 3, 3, 5, 10, 5, 7, 25, 25, 7, 10, 52, 87, 52, 10, 14, 98, 245, 245, 98, 14, 18, 168, 588, 882, 588, 168, 18, 22, 270, 1260, 2646, 2646, 1260, 270, 22, 27, 412, 2475, 6930, 9702, 6930, 2475, 412, 27, 33, 605, 4537, 16335, 30492, 30492, 16335, 4537, 605, 33

Offset: 1

Author

Roger L. Bagula, Mar 07 2010

Keywords

• nonn
• tabl

Comments

Row sums are {1, 6, 20, 64, 211, 714, 2430, 8396, 29390, 104004, 371448, 1337216, ...}.

Examples

			Triangle begins as:
   1;
   3,   3;
   5,  10,    5;
   7,  25,   25,    7;
  10,  52,   87,   52,   10;
  14,  98,  245,  245,   98,   14;
  18, 168,  588,  882,  588,  168,   18;
  22, 270, 1260, 2646, 2646, 1260,  270,  22;
  27, 412, 2475, 6930, 9702, 6930, 2475, 412, 27;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A166454.

Cf. A011848 (right diagonal).

Programs

Magma

[[Floor(Binomial(n+1, k)*Binomial(n+2, k)/(2*k+2)): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Apr 13 2019

Mathematica

T[n_, k_] = Floor[Binomial[n+1, k]*Binomial[n+2, k]/(2*(k+1))];
Table[T[n, k], {n,1,12}, {k,1,n}]//Flatten (* modified by G. C. Greubel, Apr 13 2019 *)

PARI

{T(n,k) = (binomial(n+1,k)*binomial(n+2,k)/(2*k+2))\1};
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 13 2019

Sage

[[floor(binomial(n+1,k)*binomial(n+2,k)/(2*k+2)) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Apr 13 2019

Formula

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

Extensions

Partially edited by Jon E. Schoenfield, Dec 02 2013

Edited by G. C. Greubel, Apr 13 2019