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

1,2

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.

Links

Table of n, a(n) for n=1..44.

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.

Example

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;

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

Crossrefs

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

Sequence in context: A062991 A234950 A275228 * A073474 A067311 A341487

Adjacent sequences: A118981 A118982 A118983 * A118985 A118986 A118987

Keyword

nonn,tabl

Author

Alford Arnold, May 07 2006

Extensions

Edited by R. J. Mathar, May 19 2016

Status

approved