A323849 - OEIS

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).

%I #22 Feb 11 2019 15:26:01

%S 1,1,4,1,1,18,44,18,1,1,68,615,1236,615,68,1,1,250,7313,46812,84910,

%T 46812,7313,250,1,1,922,85801,1592348,8241540,14024408,8241540,

%U 1592348,85801,922,1,1,3430,1030330,54926890,759337545,3397542544,5530983756,3397542544,759337545,54926890,1030330,3430,1

%N 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).

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

%H Alois P. Heinz, <a href="/A323849/b323849.txt">Rows n = 1..15, flattened</a>

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

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

%e Triangle begins:

%e n\d 0 1 2 3 4 5 6 7 8 9 10

%e 1 1

%e 2 1 4 1

%e 3 1 18 44 18 1

%e 4 1 68 615 1236 615 68 1

%e 5 1 250 7313 46812 84910 46812 7313 250 1

%e 6 1 922 85801 1592348 8241540 14024408 8241540 1592348 85801 922 1

%e ...

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

%Y T(n,n-1) gives A306372.

%Y Cf. A323848.

%K nonn,tabf

%O 1,3

%A _N. J. A. Sloane_, Feb 07 2019

%E Edited by _Alois P. Heinz_, Feb 11 2019