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A372973
Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).
0
1, 1, 1, 2, 1, 1, 6, 2, 3, 1, 24, 6, 11, 6, 1, 120, 24, 50, 35, 10, 1, 720, 120, 274, 225, 85, 15, 1, 5040, 720, 1764, 1624, 735, 175, 21, 1, 40320, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 362880, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1
OFFSET
0,4
LINKS
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 6.
FORMULA
T(n,0) = n!; T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] log(1/(1-x))^(k-1)/(1-x).
T(n,1) = (n-1)! for n > 0.
T(n,2) = A000254(n-1) for n > 1.
EXAMPLE
The triangle begins:
1;
1, 1;
2, 1, 1;
6, 2, 3, 1;
24, 6, 11, 6, 1;
120, 24, 50, 35, 10, 1;
720, 120, 274, 225, 85, 15, 1;
...
MATHEMATICA
T[n_, 0]:=n!; T[n_, k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)Log[1/(1-x)]^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}]//Flatten
CROSSREFS
Cf. A000012 (right diagonal), A000254, A000399 (k=3), A000454 (k=4), A000482 (k=5), A001233 (k=6), A001234 (k=7), A098558 (row sums), A179865 (subdiagonal), A243569 (k=8), A243570 (k=9).
Triangle A130534 with 1st column A000142.
Sequence in context: A179972 A085826 A112477 * A156984 A181621 A307070
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 26 2024
STATUS
approved