OFFSET
0,5
COMMENTS
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).
T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1).
Row sums = 2^n*binomial(n,2) = A100381(n).
REFERENCES
J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2.
FORMULA
E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x).
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)!).
T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ).
EXAMPLE
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, 225, 90, 15;
21, 147, 441, 735, 735, 441, 147, 21;
28, 224, 784, 1568, 1960, 1568, 784, 224, 28;
36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36;
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 12 2014
STATUS
approved