cp's OEIS Frontend

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

%I A144404 #18 Mar 27 2021 23:52:17
%S A144404 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 A144404 659,101,1,1,139,1301,3639,3639,1301,139,1,1,183,2323,9351,14629,9351,
%U A144404 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 A144404 Triangle T(n,k) = 3*binomial(n, k)^2 - binomial(n, k) - 1, read by rows.
%H A144404 G. C. Greubel, <a href="/A144404/b144404.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A144404 From _Robert Israel_, Jul 11 2016: (Start)
%F A144404 T(n,m) = A144390(A007318(n,m)) = 3*A008459(n,m) - A007318(n,m).
%F A144404 Row sums: 3*binomial(2*n,n) - 2^n - n - 1.
%F A144404 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 A144404 Triangle begins as:
%e A144404   1;
%e A144404   1,   1;
%e A144404   1,   9,    1;
%e A144404   1,  23,   23,     1;
%e A144404   1,  43,  101,    43,      1;
%e A144404   1,  69,  289,   289,     69,      1;
%e A144404   1, 101,  659,  1179,    659,    101,      1;
%e A144404   1, 139, 1301,  3639,   3639,   1301,    139,     1;
%e A144404   1, 183, 2323,  9351,  14629,   9351,   2323,   183,    1;
%e A144404   1, 233, 3851, 21083,  47501,  47501,  21083,  3851,  233,   1;
%e A144404   1, 289, 6029, 43079, 132089, 190259, 132089, 43079, 6029, 289, 1;
%p A144404 T:= (n,m) -> 3*Binomial(n,m)^2 - Binomial(n,m)-1:
%p A144404 seq(seq(T(n,m),m=0..n),n=0..10); # _Robert Israel_, Jul 11 2016
%t A144404 Table[3*Binomial[n,k]^2 -Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten
%o A144404 (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 A144404 (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 A144404 Cf. A000984, A007318, A008459, A144390, A144405.
%K A144404 nonn,tabl
%O A144404 0,5
%A A144404 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 03 2008
%E A144404 Offset changed by _Robert Israel_, Jul 11 2016