A000399 -id:A000399 - cp's OEIS frontend

A126676 Product_{i=3..n} |Stirling_1(i,3)|.

1, 6, 210, 47250, 76734000, 1007670888000, 119030115974112000, 139586617002841142400000, 1780228528528626725525222400000, 268668486342998603475774101557862400000, 518949181117200500866291866458307199421644800000, 13802344536617079696362247660105733403745253587733708800000

Offset: 3

N. J. A. Sloane, Feb 13 2007

nonn

Partial products of A000399.

FoldList[Times,StirlingS1[Range[3,20],3]]//Abs (* Harvey P. Dale, Jul 21 2018 *)

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

Original entry on oeis.org

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

Stefano Spezia, May 26 2024

			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;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 6.

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.

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

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.

cp's OEIS Frontend

A126676 Product_{i=3..n} |Stirling_1(i,3)|.

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