A373173 Triangle read by rows: the exponential almost-Riordan array ( exp(exp(x)-1) | exp(x), exp(x)-1 ).

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, 877, 1, 63, 301, 350, 140, 21, 1, 4140, 1, 127, 966, 1701, 1050, 266, 28, 1, 21147, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 115975, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1

Offset: 0

Author

Stefano Spezia, May 26 2024

Keywords

• nonn
• tabl

Examples

			The triangle begins:
    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;
  ...
		

Links

• Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 14.

Crossrefs

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).

Triangle A008277 with 1st column A000110.

Programs

Mathematica

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

Formula

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).

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