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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
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Werner Schulte, Aug 02 2020
STATUS
approved