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A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

%I A237765 #10 Feb 13 2014 18:45:33
%S A237765 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 A237765 225,90,15,21,147,441,735,735,441,147,21,28,224,784,1568,1960,1568,
%U A237765 784,224,28,36,324,1296,3024,4536,4536,3024,1296,324,36
%N A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
%C A237765 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 A237765 T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1).
%C A237765 Row sums = 2^n*binomial(n,2) = A100381(n).
%D A237765 J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2.
%F A237765 E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x).
%F A237765 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 A237765 T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ).
%e A237765 0;
%e A237765 0,  0;
%e A237765 1,  2,   1;
%e A237765 3,  9,   9,    3;
%e A237765 6,  24,  36,   24,   6;
%e A237765 10, 50,  100,  100,  50,   10;
%e A237765 15, 90,  225,  300,  225,  90,   15;
%e A237765 21, 147, 441,  735,  735,  441,  147,  21;
%e A237765 28, 224, 784,  1568, 1960, 1568, 784,  224,  28;
%e A237765 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36;
%t A237765 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 A237765 Cf. A134400.
%K A237765 nonn,tabl
%O A237765 0,5
%A A237765 _Geoffrey Critzer_, Feb 12 2014