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

%I

%S 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,

%T 659,101,1,1,139,1301,3639,3639,1301,139,1,1,183,2323,9351,14629,9351,

%U 2323,183,1,1,233,3851,21083,47501,47501,21083,3851,233,1,1,289,6029,43079,132089,190259,132089,43079,6029,289,1

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

%H G. C. Greubel, <a href="/A144404/b144404.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _Robert Israel_, Jul 11 2016: (Start)

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

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

%F 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)

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 9, 1;

%e 1, 23, 23, 1;

%e 1, 43, 101, 43, 1;

%e 1, 69, 289, 289, 69, 1;

%e 1, 101, 659, 1179, 659, 101, 1;

%e 1, 139, 1301, 3639, 3639, 1301, 139, 1;

%e 1, 183, 2323, 9351, 14629, 9351, 2323, 183, 1;

%e 1, 233, 3851, 21083, 47501, 47501, 21083, 3851, 233, 1;

%e 1, 289, 6029, 43079, 132089, 190259, 132089, 43079, 6029, 289, 1;

%p T:= (n,m) -> 3*Binomial(n,m)^2 - Binomial(n,m)-1:

%p seq(seq(T(n,m),m=0..n),n=0..10); # _Robert Israel_, Jul 11 2016

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

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

%o (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

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

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008

%E Offset changed by _Robert Israel_, Jul 11 2016

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Last modified July 30 07:58 EDT 2021. Contains 346348 sequences. (Running on oeis4.)