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A336598
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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 cross the marked chord.
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5
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1, 4, 2, 21, 18, 6, 144, 156, 96, 24, 1245, 1500, 1260, 600, 120, 13140, 16470, 16560, 11160, 4320, 720, 164745, 207270, 231210, 194040, 108360, 35280, 5040, 2399040, 2976120, 3507840, 3402000, 2419200, 1149120, 322560, 40320
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OFFSET
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1,2
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LINKS
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FORMULA
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T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.
E.g.f.: x/sqrt(1 - 2*x)/(1 - x*(1 + y)).
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EXAMPLE
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Triangle begins:
1;
4, 2;
21, 18, 6;
144, 156, 96, 24;
1245, 1500, 1260, 600, 120;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.
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MATHEMATICA
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CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)), {x, 0, 10}]]/.{x^n_.->x^n*n!}, {x, y}]
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PROG
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(PARI)
T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jul 29 2020
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CROSSREFS
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Row sums are n*A001147(n) for n > 0.
First column is A233481(n) for n > 0.
Leading diagonal is A000142(n) for n > 0.
Sub-leading diagonal is n*A000142(n) for n > 1.
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KEYWORD
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AUTHOR
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STATUS
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approved
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