A329854 Triangle read by rows: T(n,k) = ((n - k)*(n + k - 1) + 2)/2, 0 <= k <= n.

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 7, 6, 4, 1, 11, 11, 10, 8, 5, 1, 16, 16, 15, 13, 10, 6, 1, 22, 22, 21, 19, 16, 12, 7, 1, 29, 29, 28, 26, 23, 19, 14, 8, 1, 37, 37, 36, 34, 31, 27, 22, 16, 9, 1, 46, 46, 45, 43, 40, 36, 31, 25, 18, 10, 1, 56, 56, 55, 53, 50, 46, 41, 35, 28, 20, 11, 1

Offset: 0

Author

Werner Schulte, Nov 22 2019

Keywords

• nonn
• tabl

Comments

This triangle equals A309559 with reversed rows and supplemented main diagonal (all terms are 1).

There are two lower triangular matrices M and N so that the matrix product M * N equals T (seen as a matrix).

/ 1 \ / 1 \

| 0 1 | | 1 1 |

| 0 1 1 | | 1 1 1 |

M(n,k) = | 0 1 2 1 | N(n,k) = | 1 1 1 1 |

| 0 1 2 3 1 | | 1 1 1 1 1 |

| 0 1 2 3 4 1 | | 1 1 1 1 1 1 |

\ . . . . . . . / \ . . . . . . . /

The matrix product N * M equals the rascal triangle A077028 (seen as a matrix).

Examples

			The triangle T(n,k) starts:
n \ k :   0    1    2    3    4    5    6    7    8    9   10   11
==================================================================
   0  :   1
   1  :   1    1
   2  :   2    2    1
   3  :   4    4    3    1
   4  :   7    7    6    4    1
   5  :  11   11   10    8    5    1
   6  :  16   16   15   13   10    6    1
   7  :  22   22   21   19   16   12    7    1
   8  :  29   29   28   26   23   19   14    8    1
   9  :  37   37   36   34   31   27   22   16    9    1
  10  :  46   46   45   43   40   36   31   25   18   10    1
  11  :  56   56   55   53   50   46   41   35   28   20   11    1
etc.
		

Crossrefs

Row sums equal A116731(n+1).

Row sums apart from column 0 equal A081489.

Cf. A077028, A309559.

Formula

O.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((t^2+(1-t)^2) * (1-x*t) + x * t^2 * (1-t)) / ((1-t)^3 * (1-x*t)^2).

G.f. of column k: Sum_{n>=k} T(n,k) * t^n = t^k * (t^2/(1-t)^3 + 1/(1-t) + k*t/(1-t)^2) for k >= 0.

T(n,k) = 1 + T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 0 < k < n with initial values T(n,0) = (n*(n-1)+2)/2 and T(n,n) = 1 for n >= 0.

T(n,k) = (2 + T(n-1,k-1) * T(n-1,k+1)) / T(n-2,k) for 0 < k < n-1 with initial values given above and T(n,n-1) = n for n > 0.

Referring to the triangle M(n,k) (see comments), we get:

(1) Sum_{k=0..n} (k+1) * M(n,k) = A116731(n+1) for n >= 0;

(2) Sum_{k=1..n} k * M(n,k) = A081489(n) for n >= 1.

T(n,k) = T(n-1,k-1) + n-k for 0 < k <= n with initial values T(n,0) = (n*(n-1)+2)/2 for n >= 0.

T(n,k) = 2 * T(n-1,k-1) - T(n-2,k-2) for 1 < k <= n with initial values T(0,0) = 1 and T(n,0) = T(n,1) = (n*(n-1)+2)/2 for n > 0.