A118984 Triangular T(n,k) which contains in column k >= 0 the elements of the Stirling transform of the unsigned sequence Stirling1(j+k,j), j >= 0.

1, 2, 1, 5, 6, 2, 15, 31, 23, 6, 52, 160, 195, 110, 24, 203, 856, 1505, 1365, 634, 120, 877, 4802, 11312, 14560, 10738, 4284, 720, 4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040, 21147, 175896, 652703, 1404186, 1908249, 1672524, 921212, 291024

Offset: 1

Author

Alford Arnold, May 07 2006

Keywords

• nonn
• tabl

Comments

The initial array of unsigned Stirling numbers of the first kind (filled with an appropriate number of zeros) starts (see A094638)

1, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 0, 0, 0, 0, 0, 0, ...

1, 3, 2, 0, 0, 0, 0, 0, ...

1, 6, 11, 6, 0, 0, 0, 0, ...

1, 10, 35, 50, 24, 0, 0, 0, ...

1, 15, 85, 225, 274, 120, 0, 0, ...

1, 21, 175, 735, 1624, 1764, 720, 0, ...

1, 28, 322, 1960, 6769, 13132, 13068, 5040, ...

The Stirling transform is then applied on each individual column. - R. J. Mathar, May 19 2016.

Examples

			The array begins
     1;
     2,     1;
     5,     6,     2;
    15,    31,    23,      6;
    52,   160,   195,    110,     24;
   203,   856,  1505,   1365,    634,   120;
   877,  4802, 11312,  14560,  10738,  4284,   720;
  4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040;
		

Links

• Sela Fried, The expected degree of noninvertibility of compositions of functions and a related combinatorial identity, arXiv:2202.13061 [math.CO], 2022. See Corollary 2.6 for a combinatorial identity of a signed version of this sequence.

Crossrefs

Cf. A000110 (first column), A000142 (diagonal), A000670 (row sums), A003128 (2nd column), A008275, A008277.

Programs

Maple

read("transforms"):
A118984 := proc(n,k)
    [seq(0,j=0..k-2), seq( (-1)^k*combinat[stirling1](j+k,j),j=0..n)] ;
    STIRLING(%) ;
    op(n,%) ;
end proc: # R. J. Mathar, May 19 2016

Extensions

Edited by R. J. Mathar, May 19 2016