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

1,2

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A166454.

Cf. A011848 (right diagonal).

Sequence in context: A202674 A027170 A132775 * A217521 A331925 A252943

Adjacent sequences: A174099 A174100 A174101 * A174103 A174104 A174105

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 07 2010

Extensions

Partially edited by Jon E. Schoenfield, Dec 02 2013

Edited by G. C. Greubel, Apr 13 2019

Status

approved