A373173 - OEIS

%I #10 Jun 23 2024 16:14:15

%S 1,1,1,2,1,1,5,1,3,1,15,1,7,6,1,52,1,15,25,10,1,203,1,31,90,65,15,1,

%T 877,1,63,301,350,140,21,1,4140,1,127,966,1701,1050,266,28,1,21147,1,

%U 255,3025,7770,6951,2646,462,36,1,115975,1,511,9330,34105,42525,22827,5880,750,45,1

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

%H 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 14.

%F T(n,0) = n! * [x^n] exp(exp(x)-1); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] exp(x)*(exp(x)-1)^(k-1).

%F T(n,2) = A000225(n-1) for n > 1.

%e The triangle begins:

%e     1;

%e     1, 1;

%e     2, 1,  1;

%e     5, 1,  3,  1;

%e    15, 1,  7,  6,  1;

%e    52, 1, 15, 25, 10,  1;

%e   203, 1, 31, 90, 65, 15, 1;

%e   ...

%t T[n_,0]:=n!SeriesCoefficient[Exp[Exp[x]-1],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[Exp[x](Exp[x]-1)^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

%Y Cf. A000012 (k=1), A000225, A000392 (k=3), A000453 (k=4), A000481 (k=5), A000770 (k=6), A000771 (k=7), A049394 (k=8), A049435 (k=10), A049447 (k=9).

%Y Triangle A008277 with 1st column A000110.

%K nonn,tabl

%O 0,4

%A _Stefano Spezia_, May 26 2024