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A144404
Triangle T(n,k) = 3*binomial(n, k)^2 - binomial(n, k) - 1, read by rows.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Offset changed by Robert Israel, Jul 11 2016
STATUS
approved