A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).

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

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

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

Adjacent sequences: A372970 A372971 A372972 * A372974 A372975 A372976

Keyword

nonn,tabl

Author

Stefano Spezia, May 26 2024

Status

approved