A323849 -id:A323849 - cp's OEIS frontend

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

N. J. A. Sloane, Feb 07 2019

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

			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
...

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

Alois P. Heinz, Rows n = 1..14, flattened
Don Knuth, A conjecture about noncrossing paths, Feb 06 2019.

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

Cf. A005157, A323849.

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

More terms from Alois P. Heinz, Feb 07 2019

A252869 Number of n X n nonnegative integer arrays with upper left 0 and lower right 2n-4 and value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

0, 1, 44, 615, 7313, 85801, 1030330, 12742873, 161937617, 2105918045, 27918122131, 376111502977, 5136280530303, 70961632168027, 990271171218092, 13940676224604419, 197768060575469207, 2824828302263709187, 40595088350375470407, 586585037283553891311

Offset: 1

R. H. Hardin, Dec 24 2014

nonn

			Some solutions for n=3
..0..0..1....0..0..0....0..0..1....0..1..1....0..1..1....0..1..2....0..1..2
..1..1..1....0..0..1....0..0..1....1..1..2....0..1..1....0..1..2....1..2..2
..1..2..2....1..1..2....0..1..2....1..2..2....1..2..2....1..1..2....2..2..2

Main diagonal of A252876.

Column d=2 of A323849.

a(16)-a(20) from Alois P. Heinz, Feb 07 2019

A306372 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=1, m_{n,n}=n, and all rows and columns are (weakly) monotonic without jumps larger than 1.

Original entry on oeis.org

1, 4, 44, 1236, 84910, 14024408, 5530983756, 5192169001644, 11583105980431652, 61353264333274642456, 771168645274178148758934, 22993716372161876140519972488, 1625987621134180524757435478838056, 272649922373702777601949547955204462396, 108399145705124270227889424469312592196654704

Offset: 1

Alois P. Heinz, Feb 11 2019

nonn

Cf. A323849, A367432.

a(n) = A323849(n,n-1).

cp's OEIS Frontend

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).

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A252869 Number of n X n nonnegative integer arrays with upper left 0 and lower right 2n-4 and value increasing by 0 or 1 with every step right or down.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A306372 Number of n X n integer matrices (m_{i,j}) such that m_{1,1}=1, m_{n,n}=n, and all rows and columns are (weakly) monotonic without jumps larger than 1.

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