A323848 Irregular triangle read by rows: T(n,d) (n >= 1, d <= n-1 for n>1) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1), and M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

0, 4, 18, 25, 68, 386, 256, 250, 4657, 12200, 4356, 922, 54219, 432842, 608993, 123904, 3430, 642815, 14697256, 60650883, 49489706, 5909761, 12868, 7852836, 514608568, 5713126349, 13458882036, 6648891794, 473497600, 48618, 98755951, 18971384148, 558848240787, 3406380649146, 4857082197177, 1489334202216, 63799687396

Offset

1,2

Comments

T(n,n-1) = A005157(n-1)^2 for n >= 2. See Knuth (2019) link.

References

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

Links

Alois P. Heinz, Rows n = 1..14, flattened

Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.

Formula

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

Example

Triangle begins:
  n\d    1      2        3        4        5       6  7
   1     0      0        0        0        0       0  0
   2     4      0        0        0        0       0  0
   3    18     25        0        0        0       0  0
   4    68    386      256        0        0       0  0
   5   250   4657    12200     4356        0       0  0
   6   922  54219   432842   608993   123904       0  0
   7  3430 642815 14697256 60650883 49489706 5909761  0
...

Crossrefs

Columns d=1-2 give: A115112, A306322.

Cf. A005157, A323849.

Sequence in context: A192195 A099565 A063563 * A358375 A166749 A370406

Adjacent sequences: A323845 A323846 A323847 * A323849 A323850 A323851

Keyword

nonn,tabf

Author

N. J. A. Sloane, Feb 07 2019

Extensions

More terms from Alois P. Heinz, Feb 07 2019

Status

approved