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A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

%I A373168 #13 Jul 28 2024 03:56:46
%S A373168 1,1,1,3,1,1,13,2,2,1,73,6,6,3,1,501,24,24,12,4,1,4051,120,120,60,20,
%T A373168 5,1,37633,720,720,360,120,30,6,1,394353,5040,5040,2520,840,210,42,7,
%U A373168 1,4596553,40320,40320,20160,6720,1680,336,56,8,1,58941091,362880,362880,181440,60480,15120,3024,504,72,9,1
%N A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).
%H A373168 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 13.
%F A373168 T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).
%F A373168 T(n,3) = A001710(n-1) for n > 2.
%F A373168 T(n,4) = A001715(n-1) for n > 3.
%F A373168 T(n,5) = A001720(n-1) for n > 4.
%F A373168 T(n,6) = A001725(n-1) for n > 5.
%F A373168 T(n,7) = A001730(n-1) for n > 6.
%F A373168 T(n,8) = A049388(n-8) for n > 7.
%F A373168 T(n,9) = A049389(n-9) for n > 8.
%F A373168 T(n,10) = A049398(n-10) for n > 9.
%F A373168 T(n,11) = A051431(n-11) for n > 10.
%e A373168 The triangle begins:
%e A373168      1;
%e A373168      1,   1;
%e A373168      3,   1,   1;
%e A373168     13,   2,   2,  1;
%e A373168     73,   6,   6,  3,  1;
%e A373168    501,  24,  24, 12,  4, 1;
%e A373168   4051, 120, 120, 60, 20, 5, 1;
%e A373168   ...
%t A373168 T[n_,0]:=n!SeriesCoefficient[Exp[x/(1-x)],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten
%Y A373168 Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431.
%Y A373168 Triangle A094587 with 1st column A000262.
%K A373168 nonn,tabl
%O A373168 0,4
%A A373168 _Stefano Spezia_, May 26 2024