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

%I A372973 #42 May 27 2024 15:47:11
%S A372973 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,
%T A372973 15,1,5040,720,1764,1624,735,175,21,1,40320,5040,13068,13132,6769,
%U A372973 1960,322,28,1,362880,40320,109584,118124,67284,22449,4536,546,36,1
%N A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).
%H A372973 Y. Alp and E. G. Kocer, <a href="https://doi.org/10.1007/s00025-024-02193-5">Exponential Almost-Riordan Arrays</a>, Results Math 79, 173 (2024). See page 6.
%F A372973 T(n,0) = n!; T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] log(1/(1-x))^(k-1)/(1-x).
%F A372973 T(n,1) = (n-1)! for n > 0.
%F A372973 T(n,2) = A000254(n-1) for n > 1.
%e A372973 The triangle begins:
%e A372973     1;
%e A372973     1,   1;
%e A372973     2,   1,   1;
%e A372973     6,   2,   3,   1;
%e A372973    24,   6,  11,   6,  1;
%e A372973   120,  24,  50,  35, 10,  1;
%e A372973   720, 120, 274, 225, 85, 15, 1;
%e A372973   ...
%t A372973 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
%Y A372973 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).
%Y A372973 Triangle A130534 with 1st column A000142.
%K A372973 nonn,tabl
%O A372973 0,4
%A A372973 _Stefano Spezia_, May 26 2024