A308498 Triangle read by rows where T(n,k), n>=1, 1<=k<=n is the number of (0,1)-matrices of size n with the first row and column sum = k and remaining sums = 1.

1, 2, 1, 6, 5, 1, 24, 27, 10, 1, 120, 168, 88, 17, 1, 720, 1200, 800, 225, 26, 1, 5040, 9720, 7800, 2850, 486, 37, 1, 40320, 88200, 82320, 36750, 8232, 931, 50, 1, 362880, 887040, 940800, 493920, 136416, 20384, 1632, 65, 1, 3628800, 9797760, 11612160, 6985440

Offset

1,2

Links

Table of n, a(n) for n=1..49.

Formula

T(n,k) = ((n-1)!)^2 * (k^2+n-k) / ((k!)^2 * (n-k)!).

T(n,1) = A000142(n).

T(n,2) = A138772(n).

T(n,n-1) = A002522(n-1).

T(n,n) = 1.

Example

For n=4, k=3:
1110  1101  1011  1110  1101  1011  1110  1101  1011  0111
1000  1000  1000  1000  1000  1000  0001  0010  0100  1000
1000  1000  1000  0001  0010  0100  1000  1000  1000  1000
0001  0010  0100  1000  1000  1000  1000  1000  1000  1000
so T(4,3)=10.
Triangle begins:
1
2,1
6,5,1
24,27,10,1
120,168,88,17,1
720,1200,800,225,26,1
5040,9720,7800,2850,486,37,1
40320,88200,82320,36750,8232,931,50,1
362880,887040,940800,493920,136416,20384,1632,65,1
3628800,9797760,11612160,6985440,2286144,423360,44928,2673,82,1
39916800,117936000,154224000,104328000,39372480,8678880,1144800,90450,4150,101,1

Crossrefs

Cf. A000142, A002522, A138772.

Sequence in context: A049444 A136124 A143491 * A295517 A070918 A113381

Adjacent sequences: A308495 A308496 A308497 * A308499 A308500 A308501

Keyword

nonn,tabl

Author

Lars Blomberg, Jun 01 2019

Status

approved