A144404 Triangle T(n,k) = 3*binomial(n, k)^2 - binomial(n, k) - 1, read by rows.

1, 1, 1, 1, 9, 1, 1, 23, 23, 1, 1, 43, 101, 43, 1, 1, 69, 289, 289, 69, 1, 1, 101, 659, 1179, 659, 101, 1, 1, 139, 1301, 3639, 3639, 1301, 139, 1, 1, 183, 2323, 9351, 14629, 9351, 2323, 183, 1, 1, 233, 3851, 21083, 47501, 47501, 21083, 3851, 233, 1, 1, 289, 6029, 43079, 132089, 190259, 132089, 43079, 6029, 289, 1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   9,    1;
  1,  23,   23,     1;
  1,  43,  101,    43,      1;
  1,  69,  289,   289,     69,      1;
  1, 101,  659,  1179,    659,    101,      1;
  1, 139, 1301,  3639,   3639,   1301,    139,     1;
  1, 183, 2323,  9351,  14629,   9351,   2323,   183,    1;
  1, 233, 3851, 21083,  47501,  47501,  21083,  3851,  233,   1;
  1, 289, 6029, 43079, 132089, 190259, 132089, 43079, 6029, 289, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000984, A007318, A008459, A144390, A144405.

Programs

Magma

[3*Binomial(n, k)^2 -Binomial(n, k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2021

Maple

T:= (n,m) -> 3*Binomial(n,m)^2 - Binomial(n,m)-1:
seq(seq(T(n,m),m=0..n),n=0..10); # Robert Israel, Jul 11 2016

Mathematica

Table[3*Binomial[n,k]^2 -Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten

Sage

flatten([[3*binomial(n, k)^2 -binomial(n, k) -1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 27 2021

Formula

From Robert Israel, Jul 11 2016: (Start)

T(n,m) = A144390(A007318(n,m)) = 3*A008459(n,m) - A007318(n,m).

Row sums: 3*binomial(2*n,n) - 2^n - n - 1.

G.f. as triangle: g(x,y) = 3/sqrt(1-2*x-2*x*y+x^2-2*x^2*y+x^2*y^2) - 1/(1-x-x*y)+1/((1-x)*(1-x*y)). (End)

Extensions

Offset changed by Robert Israel, Jul 11 2016