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A336601
Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.
5
1, 4, 2, 22, 16, 7, 160, 136, 88, 36, 1464, 1344, 1044, 624, 249, 16224, 15504, 13344, 9624, 5484, 2190, 211632, 206592, 188952, 152832, 104322, 58080, 23535, 3179520, 3139200, 2977920, 2594880, 1990080, 1309680, 725040, 299880, 54092160, 53729280, 52096320, 47681280, 39652560, 29174400, 18809640, 10473120, 4426065
OFFSET
1,2
LINKS
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.
FORMULA
E.g.f.: arctan(x*(1 - y)/sqrt((1 - 2*x)*(1 - 2*x*y)))/(1 - y)/sqrt(1 - 2*x).
EXAMPLE
Triangle begins:
1;
4, 2;
22, 16, 7;
160, 136, 88, 36;
1464, 1344, 1044, 624, 249;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.
MATHEMATICA
CoefficientList[Normal[Series[1/(1-y)/Sqrt[1-2*x]*ArcTan[(x*(1-y))/Sqrt[(1-2*x)]/Sqrt[1-2*y*x]], {x, 0, 10}]]/.{x^n_.->x^n*n!}, {x, y}]
CROSSREFS
Row sums are n*A001147(n) for n > 0.
The first column is A087547(n) for n > 0.
Leading diagonal is A034430(n-1) for n > 0.
Sequence in context: A100400 A285439 A336598 * A241437 A030211 A134461
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Jul 31 2020
STATUS
approved