login
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).
3
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, 10, 35, 50, 24, 1, 11, 41, 61, 30, 1, 12, 49, 78, 40, 1, 13, 59, 107, 60, 1, 14, 71, 154, 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
OFFSET
1,5
COMMENTS
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.
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);
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.
EXAMPLE
Table begins:
+---+
| 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 10 35 50 24 |
| 1 11 41 61 30 |
| 1 12 49 78 40 |
| 1 13 59 107 60 |
| 1 14 71 154 120 |
+-----------------------+----+
| 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 |
+----------------------------+----+
| 1 21 175 735 1624 1764 720 |
| 1 22 190 820 1849 2038 840 |
| 1 23 207 925 2144 2412 1008 |
| 1 24 226 1056 2545 2952 1260 |
| 1 25 247 1219 3112 3796 1680 |
| 1 26 270 1420 3929 5274 2520 |
| 1 27 295 1665 5104 8028 5040 |
+---------------------------------+----+
etc.
PROG
(SWI-Prolog) See link.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Luc Rousseau, Jun 10 2020
STATUS
approved