A237765 - OEIS

%I #10 Feb 13 2014 18:45:33

%S 0,0,0,1,2,1,3,9,9,3,6,24,36,24,6,10,50,100,100,50,10,15,90,225,300,

%T 225,90,15,21,147,441,735,735,441,147,21,28,224,784,1568,1960,1568,

%U 784,224,28,36,324,1296,3024,4536,4536,3024,1296,324,36

%N Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

%C T(n,k) is the number of ways to underline exactly two elements of {1,2,...,n} and then circle exactly k elements. (The k elements that are circled are not necessarily different from the two underlined elements).

%C T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1).

%C Row sums = 2^n*binomial(n,2) = A100381(n).

%D J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2.

%F E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x).

%F E.g.f. for column k: x^2/2!*exp(x)*(x^k/k! + 2*x^(k-1)/(k-1)! + x^(k-2)/(k-2)!).

%F T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ).

%e 0;

%e 0,  0;

%e 1,  2,   1;

%e 3,  9,   9,    3;

%e 6,  24,  36,   24,   6;

%e 10, 50,  100,  100,  50,   10;

%e 15, 90,  225,  300,  225,  90,   15;

%e 21, 147, 441,  735,  735,  441,  147,  21;

%e 28, 224, 784,  1568, 1960, 1568, 784,  224,  28;

%e 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36;

%t Table[Table[Binomial[n,2](Binomial[n-2,r]+2Binomial[n-2,r-1]+Binomial[n-2,r-2]),{r,0,n}],{n,0,9}]//Grid

%Y Cf. A134400.

%K nonn,tabl

%O 0,5

%A _Geoffrey Critzer_, Feb 12 2014