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

0,4

Links

Table of n, a(n) for n=0..54.

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).

Example

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).

Sequence in context: A224887 A151571 A193957 * A209769 A114230 A209753

Adjacent sequences: A336743 A336744 A336745 * A336747 A336748 A336749

Keyword

nonn,easy,tabl

Author

Werner Schulte, Aug 02 2020

Status

approved