A062154 - OEIS

%I #14 Aug 18 2024 02:01:02

%S 1,0,2,1,0,1,13,18,6,0,0,18,189,450,360,90,0,0,6,450,4842,16380,22140,

%T 12600,2520,0,0,0,360,16380,190080,832950,1631700,1537200,680400,

%U 113400,0,0,0,90,22140,832950,10520010,56609280,147533400,200377800

%N Number T(n,m) of n X m matrices over {0,1,2} with all row and column sums equal to 1 or 2, m=0,..,2*n.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(Problem 3.4.15).

%H Andrew Howroyd, <a href="/A062154/b062154.txt">Table of n, a(n) for n = 0..960</a> (rows 0..30)

%F Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1/sqrt(1-x*y)*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2)).

%F Sum_{n >= 0, m >= 0} T(n, m)*x^n/n!*y^m/m! = 1+(1/2*y^2+2*y)*x+(1/8*y^4+3/2*y^3+13/4*y^2+1/2*y)*x^2+(1/48*y^6+1/2*y^5+25/8*y^4+21/4*y^3+3/2*y^2)*x^3+...

%e Triangle begins:

%e [0]  1;

%e [1]  0, 2, 1;

%e [2]  0, 1, 13, 18, 6;

%e [3]  0, 0, 18, 189, 450, 360, 90;

%e [4]  0, 0, 6, 450, 4842, 16380, 22140, 12600, 2520;

%e [5]  0, 0, 0, 360, 16380, 190080, 832950, 1631700, 1537200, 680400, 113400;

%e [6]  0, 0, 0, 90, 22140, 832950, 10520010, 56609280, 147533400, 200377800, 144585000, 52390800, 7484400;

%e T(2, 2)=13, i.e. there are 13 2 X 2 matrices over {0, 1, 2} with all row and column sums equal to 1 or 2: [0 1 / 0 1], [0 1 / 0 2], [0 2 / 1 0], [1 0 / 1 0], [1 1 / 1 1], [1 1 / 2 0], [2 0 / 1 0], [1 1 / 2 0], [1 0 / 2 0], [0 1 / 0 2], [1 1 / 0 1], [1 0 / 1 1], [0 1 / 0 2].

%o (PARI)

%o Row(n)={Vecrev(serlaplace(n!*polcoef((1/sqrt(1-x*y + O(x*x^n))*exp(x*y/2+1/(1-x*y)*(x*y+x^2*y/2+x*y^2/2) + O(x*x^n))), n)))}

%o { for(n=0, 6, print(Row(n))) } \\ _Andrew Howroyd_, Feb 03 2021

%Y Row sums are A062155.

%Y Main diagonal is A062156.

%Y Final terms of each row are A000680.

%K nonn,tabf,easy

%O 0,3

%A _Vladeta Jovovic_, Jun 06 2001