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A335442 List enumerated in lexicographic order of (n, s, k), where for each n >= 1, for each s a subset of 1..n with n-1 elements, and for each k in 0..n-1, we give the value of (Sum_{t subset of s, Card(t)=k} Product_{x in t} x).

%I A335442 #15 Jan 06 2025 22:12:56
%S A335442 1,1,1,1,2,1,3,2,1,4,3,1,5,6,1,6,11,6,1,7,14,8,1,8,19,12,1,9,26,24,1,
%T A335442 10,35,50,24,1,11,41,61,30,1,12,49,78,40,1,13,59,107,60,1,14,71,154,
%U A335442 120,1,15,85,225,274,120,1,16,95,260,324,144,1,17,107,307,396,180,1,18,121,372,508,240,1,19,137,461,702,360,1,20,155,580,1044,720
%N A335442 List enumerated in lexicographic order of (n, s, k), where for each n >= 1, for each s a subset of 1..n with n-1 elements, and for each k in 0..n-1, we give the value of (Sum_{t subset of s, Card(t)=k} Product_{x in t} x).
%C A335442 This sequence can be viewed as a triangle made of square blocks of increasing sizes: 1 X 1, 2 X 2, and so on. In block number n >= 1, the bottom-right corner is n!, the top-right one (n-1)!. The first row of block number n, which is row number binomial(n, 2) if we number the rows according to their second value, is the list of the unsigned Stirling numbers of the first kind, reversed. The subsequence {last element of row n} is A077012.
%C A335442 Block number n crops up for instance when studying -log(1-r), where r = e^(i*2*Pi/n); this is Sum_{K>=1} r^K/K = Sum_{K>=0} Sum_{L=1..n} r^L/(nK+L);
%C A335442 the term for K in the first sum, if put in the same denominator and if no simplification is carried out, has a numerator which is a combination of all (nK)^I * r^J; their coefficients are precisely the elements of block number n.
%H A335442 Luc Rousseau, <a href="/A335442/a335442.pl.txt">Program for the computation of A335442 (SWI-Prolog)</a>
%e A335442 Table begins:
%e A335442   +---+
%e A335442   | 1 |
%e A335442   +---+----+
%e A335442   | 1    1 |
%e A335442   | 1    2 |
%e A335442   +--------+----+
%e A335442   | 1    3    2 |
%e A335442   | 1    4    3 |
%e A335442   | 1    5    6 |
%e A335442   +-------------+----+
%e A335442   | 1    6   11    6 |
%e A335442   | 1    7   14    8 |
%e A335442   | 1    8   19   12 |
%e A335442   | 1    9   26   24 |
%e A335442   +------------------+----+
%e A335442   | 1   10   35   50   24 |
%e A335442   | 1   11   41   61   30 |
%e A335442   | 1   12   49   78   40 |
%e A335442   | 1   13   59  107   60 |
%e A335442   | 1   14   71  154  120 |
%e A335442   +-----------------------+----+
%e A335442   | 1   15   85  225  274  120 |
%e A335442   | 1   16   95  260  324  144 |
%e A335442   | 1   17  107  307  396  180 |
%e A335442   | 1   18  121  372  508  240 |
%e A335442   | 1   19  137  461  702  360 |
%e A335442   | 1   20  155  580 1044  720 |
%e A335442   +----------------------------+----+
%e A335442   | 1   21  175  735 1624 1764  720 |
%e A335442   | 1   22  190  820 1849 2038  840 |
%e A335442   | 1   23  207  925 2144 2412 1008 |
%e A335442   | 1   24  226 1056 2545 2952 1260 |
%e A335442   | 1   25  247 1219 3112 3796 1680 |
%e A335442   | 1   26  270 1420 3929 5274 2520 |
%e A335442   | 1   27  295 1665 5104 8028 5040 |
%e A335442   +---------------------------------+----+
%e A335442   etc.
%o A335442 (SWI-Prolog) % See Rousseau link.
%Y A335442 Cf. A000142, A008275, A077012.
%K A335442 nonn,tabf
%O A335442 1,5
%A A335442 _Luc Rousseau_, Jun 10 2020