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

0,5

Links

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

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)

Example

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;

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

Prog

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

Crossrefs

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

Sequence in context: A143681 A081582 A174346 * A014761 A073702 A171822

Adjacent sequences: A144401 A144402 A144403 * A144405 A144406 A144407

Keyword

nonn,tabl

Author

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

Extensions

Offset changed by Robert Israel, Jul 11 2016

Status

approved