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A375504
Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.
1
1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 62, 39, 10, 1, 120, 396, 296, 95, 15, 1, 720, 3024, 2616, 980, 195, 21, 1, 5040, 26928, 26568, 11240, 2605, 357, 28, 1, 40320, 274320, 305892, 143464, 37290, 5971, 602, 36, 1, 362880, 3149280, 3945024, 2027460, 578514, 103824, 12292, 954, 45, 1
OFFSET
0,4
COMMENTS
In a linear chord diagram a "bubble" is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. In a crystallized diagram, all chords are either short, or "bridge" two distinct bubbles, i.e., they have one vertex in one bubble, and the other vertex in a separate bubble. T(n,k) is the number of such crystallized diagrams built from n > 0 chords, exactly k > 0 of which are short.
EXAMPLE
Triangle begins:
1;
1, 1;
2, 3, 1;
6, 12, 6, 1;
24, 62, 39, 10, 1;
120, 396, 296, 95, 15, 1;
...
For n = 3, let the vertices of the linear chord diagram be A,B,C,D,E,F. There are two diagrams with a single short chord: (AF)(BE)(CD) and (AE)(BF)(CD), and so T(3,1) = 2. There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE), and so T(3,2) = 3. Finally, there is one diagram with all three chords short: (AB)(CD)(EF), and so T(3,3)=1.
MATHEMATICA
F[n_]:=Sum[Factorial2[2*i-1]*x^i, {i, 0, n}];
T[n_, k_]:=Sum[(-1)^(n-k-l)*Factorial2[2*l-1]*Binomial[2*n-k, 2*l]*Coefficient[F[n]^(k+1), x, n-k-l], {l, 0, n-k}];
CROSSREFS
Row sums give A375505.
First column gives A000142.
The second diagonal is A000217.
Sequence in context: A135894 A335823 A247500 * A075263 A130850 A130405
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Aug 18 2024
STATUS
approved