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

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

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.

Links

Table of n, a(n) for n=0..54.

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

Cf. A134400.

Sequence in context: A249456 A234746 A359648 * A249632 A126009 A301282

Adjacent sequences: A237762 A237763 A237764 * A237766 A237767 A237768

Keyword

nonn,tabl

Author

Geoffrey Critzer, Feb 12 2014

Status

approved