A176992 Triangle T(n,m) = binomial(2n-k+1, n+1) read by rows, 0 <= k <= n.

1, 3, 1, 10, 4, 1, 35, 15, 5, 1, 126, 56, 21, 6, 1, 462, 210, 84, 28, 7, 1, 1716, 792, 330, 120, 36, 8, 1, 6435, 3003, 1287, 495, 165, 45, 9, 1, 24310, 11440, 5005, 2002, 715, 220, 55, 10, 1, 92378, 43758, 19448, 8008, 3003, 1001, 286, 66, 11, 1, 352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1

Offset: 0

Author

Roger L. Bagula, Dec 08 2010

Keywords

• nonn
• tabl
• easy

Comments

Row sums are A001791.

Obtained from A059481 by removal of the last two terms in each row, followed by row reversal.

Riordan array (c(x)/sqrt(1 - 4*x), x*c(x)) where c(x) is the g.f. of A000108. - Philippe Deléham, Jul 12 2015

Examples

			Triangle begins:
       1;
       3,      1;
      10,      4,     1;
      35,     15,     5,     1;
     126,     56,    21,     6,    1;
     462,    210,    84,    28,    7,     1;
    1716,    792,   330,   120,   36,     8,    1;
    6435,   3003,  1287,   495,   165,   45,    9,   1;
   24310,  11440,  5005,  2002,   715,  220,   55,  10,  1;
   92378,  43758, 19448,  8008,  3003, 1001,  286,  66, 11,  1;
  352716, 167960, 75582, 31824, 12376, 4368, 1365, 364, 78, 12, 1;
		

Crossrefs

Cf. A092392, A001791, A078812.

Cf. Similar triangle: A033184, A054445.

Cf. A178300 (reversal).

Programs

Magma

/* As triangle */ [[Binomial(2*n-k+1,n+1): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jul 12 2015

Maple

A176992 := proc(n,k) binomial(1+2*n-k,n+1) ; end proc: # R. J. Mathar, Dec 09 2010

Mathematica

p[t_, j_] = ((-1)^(j + 1)/2)*Sum[Binomial[k - j - 1, j + 1]*t^k, {k, 0, Infinity}];
Flatten[Table[CoefficientList[ExpandAll[p[t, j]], t], {j, 0, 10}]]

Formula

n-th row of the triangle = top row of M^n, where M is the following infinite square production matrix:

3, 1, 0, 0, 0, ...

1, 1, 1, 0, 0, ...

1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, ...

... - Philippe Deléham, Jul 12 2015