A104713 Triangle T(n,k) = binomial(n,k), read by rows, 3 <= k <=n .

1, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 35, 21, 7, 1, 56, 70, 56, 28, 8, 1, 84, 126, 126, 84, 36, 9, 1, 120, 210, 252, 210, 120, 45, 10, 1, 165, 330, 462, 462, 330, 165, 55, 11, 1, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 286, 715, 1287, 1716

Offset: 3

Author

Gary W. Adamson, Mar 19 2005

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle are:
1;
4, 1;
10, 5, 1;
20, 15, 6, 1;
35, 35, 21, 7, 1;
56, 70, 56, 28, 8, 1;
...
		

Crossrefs

Cf. A007318, A104712, A002662 (row sums).

Programs

Maple

A104713 := proc(n,k)
        binomial(n,k) ;
end proc;
seq(seq( A104713(n,k),k=3..n),n=3..16) ; # R. J. Mathar, Oct 29 2011

Formula

T(n,k) = A007318(n,k) for n>=3, 3<=k<=n.

From Peter Bala, Jul 16 2013: (Start)

The following remarks assume an offset of 0.

Riordan array (1/(1 - x)^4, x/(1 - x)).

O.g.f.: 1/(1 - t)^3*1/(1 - (1 + x)*t) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2 + ....

E.g.f.: (1/x*d/dt)^3 (exp(t)*(exp(x*t) - 1 - x*t - (x*t)^2/2!)) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2/2! + ....

The infinitesimal generator for this triangle has the sequence [4,5,6,...] on the main subdiagonal and 0's elsewhere. (End)