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%I #42 Mar 22 2022 02:54:06
%S 1,2,1,5,6,2,15,31,23,6,52,160,195,110,24,203,856,1505,1365,634,120,
%T 877,4802,11312,14560,10738,4284,720,4140,28337,85225,145096,150325,
%U 94444,33228,5040,21147,175896,652703,1404186,1908249,1672524,921212,291024
%N 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.
%C The initial array of unsigned Stirling numbers of the first kind (filled with an appropriate number of zeros) starts (see A094638)
%C 1, 0, 0, 0, 0, 0, 0, 0, ...
%C 1, 1, 0, 0, 0, 0, 0, 0, ...
%C 1, 3, 2, 0, 0, 0, 0, 0, ...
%C 1, 6, 11, 6, 0, 0, 0, 0, ...
%C 1, 10, 35, 50, 24, 0, 0, 0, ...
%C 1, 15, 85, 225, 274, 120, 0, 0, ...
%C 1, 21, 175, 735, 1624, 1764, 720, 0, ...
%C 1, 28, 322, 1960, 6769, 13132, 13068, 5040, ...
%C The Stirling transform is then applied on each individual column. - _R. J. Mathar_, May 19 2016.
%H Sela Fried, <a href="https://arxiv.org/abs/2202.13061">The expected degree of noninvertibility of compositions of functions and a related combinatorial identity</a>, arXiv:2202.13061 [math.CO], 2022. See Corollary 2.6 for a combinatorial identity of a signed version of this sequence.
%e The array begins
%e 1;
%e 2, 1;
%e 5, 6, 2;
%e 15, 31, 23, 6;
%e 52, 160, 195, 110, 24;
%e 203, 856, 1505, 1365, 634, 120;
%e 877, 4802, 11312, 14560, 10738, 4284, 720;
%e 4140, 28337, 85225, 145096, 150325, 94444, 33228, 5040;
%p read("transforms"):
%p A118984 := proc(n,k)
%p [seq(0,j=0..k-2), seq( (-1)^k*combinat[stirling1](j+k,j),j=0..n)] ;
%p STIRLING(%) ;
%p op(n,%) ;
%p end proc: # _R. J. Mathar_, May 19 2016
%Y Cf. A000110 (first column), A000142 (diagonal), A000670 (row sums), A003128 (2nd column), A008275, A008277.
%K nonn,tabl
%O 1,2
%A _Alford Arnold_, May 07 2006
%E Edited by _R. J. Mathar_, May 19 2016
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