A323849 Irregular triangle read by rows: T(n,d) (n >= 1, 0 <= d <= 2n-2) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, and M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1).

1, 1, 4, 1, 1, 18, 44, 18, 1, 1, 68, 615, 1236, 615, 68, 1, 1, 250, 7313, 46812, 84910, 46812, 7313, 250, 1, 1, 922, 85801, 1592348, 8241540, 14024408, 8241540, 1592348, 85801, 922, 1, 1, 3430, 1030330, 54926890, 759337545, 3397542544, 5530983756, 3397542544, 759337545, 54926890, 1030330, 3430, 1

Offset

1,3

References

D. E. Knuth, Email to N. J. A. Sloane, Feb 06 2019.

Links

Alois P. Heinz, Rows n = 1..15, flattened

Formula

T(n,1) = binomial(2n,n) - 2 = A115112(n).

The triangle is symmetric: T(n,d) = T(n,2n-2-d).

Example

Triangle begins:
  n\d 0   1     2       3       4        5       6       7     8   9 10
  1   1
  2   1   4     1
  3   1  18    44      18       1
  4   1  68   615    1236     615       68       1
  5   1 250  7313   46812   84910    46812    7313     250     1
  6   1 922 85801 1592348 8241540 14024408 8241540 1592348 85801 922  1
  ...

Crossrefs

Columns k=0-2 give: A000012, A115112, A252869.

T(n,n-1) gives A306372.

Cf. A323848.

Sequence in context: A203092 A139167 A211709 * A254442 A340476 A176422

Adjacent sequences: A323846 A323847 A323848 * A323850 A323851 A323852

Keyword

nonn,tabf

Author

N. J. A. Sloane, Feb 07 2019

Extensions

Edited by Alois P. Heinz, Feb 11 2019

Status

approved