A135089 Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.

1, 5, 5, 5, 10, 5, 5, 15, 15, 5, 5, 20, 30, 20, 5, 5, 25, 50, 50, 25, 5, 5, 30, 75, 100, 75, 30, 5, 5, 35, 105, 175, 175, 105, 35, 5, 5, 40, 140, 280, 350, 280, 140, 40, 5, 5, 45, 180, 420, 630, 630, 420, 180, 45, 5, 5, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 5

Offset: 0

Author

Gary W. Adamson, Nov 18 2007

Keywords

• nonn
• tabl

Comments

Row sums = A020714 (except for the first term).

Triangle T(n,k), 0 <= k <= n, read by rows given by (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 24 2013

Examples

			First few rows of the triangle:
  1;
  5,  5;
  5, 10,  5;
  5, 15, 15,   5;
  5, 20, 30   20,  5;
  5, 25, 50,  50, 25,  5;
  5, 30, 75, 100, 75, 30, 5.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows

Crossrefs

Cf. A007318, A020714, A132200, A134058, A134059.

Programs

Magma

[1] cat [5*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Mathematica

Table[5*Binomial[n,k] -4*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 22 2016; May 03 2021 *)

Sage

def A135089(n,k): return 5*binomial(n,k) - 4*bool(n==0)
flatten([[A135089(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Formula

T(n,k) = 5*binomial(n,k), n > 0, 0 <= k <= n.

Equals 2*A134059(n,k) - A007318(n,k).

G.f.: (1+4*x+4*x*y)/(1-x-x*y). - Philippe Deléham, Nov 24 2013

Sum_{k=0..n} T(n,k) = A020714(n) - 4*[n=0]. - G. C. Greubel, May 03 2021