A336746 Triangle read by rows: T(n,k) = (n-k-1+H(k+1))*((k+1)!) for 0 <= k <= n where H(k+1) = Sum_{i=0..k} 1/(i+1) for k >= 0.

0, 1, 1, 2, 3, 5, 3, 5, 11, 26, 4, 7, 17, 50, 154, 5, 9, 23, 74, 274, 1044, 6, 11, 29, 98, 394, 1764, 8028, 7, 13, 35, 122, 514, 2484, 13068, 69264, 8, 15, 41, 146, 634, 3204, 18108, 109584, 663696, 9, 17, 47, 170, 754, 3924, 23148, 149904, 1026576, 6999840

Offset: 0

Author

Werner Schulte, Aug 02 2020

Keywords

• nonn
• easy
• tabl

Examples

			The triangle starts:
n\k :  0   1   2    3    4     5      6       7        8        9
=================================================================
  0 :  0
  1 :  1   1
  2 :  2   3   5
  3 :  3   5  11   26
  4 :  4   7  17   50  154
  5 :  5   9  23   74  274  1044
  6 :  6  11  29   98  394  1764   8028
  7 :  7  13  35  122  514  2484  13068   69264
  8 :  8  15  41  146  634  3204  18108  109584   663696
  9 :  9  17  47  170  754  3924  23148  149904  1026576  6999840
...
		

Crossrefs

Cf. A001477 (column 0), A005408 (column 1), A016969 (column 2), A001705 (main diagonal), A000254 (1st subdiagonal), A000774 (2nd subdiagonal).

Formula

T(n,k) = T(n,k-1) + k * T(n-1,k-1) for 0 < k <= n with initial values T(n,0) = n for n >= 0 and T(i,j) = 0 if j < 0 or j > i.

T(n,k) = k! + T(n-1,k-1) * (k+1) for 0 < k <= n.

T(n,k) = (k+1)! + T(n-1,k) for 0 <= k < n.

E.g.f. of main diagonal (case n=0) and n-th subdiagonal (n>0): Sum_{k>=0} T(n+k,k) * x^k / k! = (n - log(1-x)) / (1-x)^2 for n >= 0.

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

G.f.: Sum_{n>=0, k=0..n} T(n,k)*x^k*y^n/k! = (y - (1-y) * log(1-x*y)) / ((1-y)^2 * (1-x*y)^2).