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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).
4
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
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.
Sequence in context: A192195 A099565 A063563 * A358375 A166749 A370406
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Feb 07 2019
EXTENSIONS
More terms from Alois P. Heinz, Feb 07 2019
STATUS
approved