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A354127
Triangle read by rows: T(n, k) is the number of graphs obtained by adding k pierced circles to a path graph P_n.
0
1, 1, 0, 2, 2, 0, 12, 10, 3, 0, 82, 82, 28, 4, 0, 646, 738, 315, 60, 5, 0, 5574, 7198, 3636, 900, 110, 6, 0, 51386, 74086, 43225, 13020, 2135, 182, 7, 0, 498026, 793490, 524784, 185920, 37940, 4452, 280, 8, 0, 5019720, 8761906, 6475959, 2634912, 642180, 95508, 8442, 408, 9, 0
OFFSET
0,4
LINKS
Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022.
FORMULA
T(n, k) = Sum_{m=k..n} (-1)^(m+k)*binomial(m, k)*O(m, n), with O(k, s) = binomial(2*s-k-1, k)*C(s-k)^2 (see Lemma 3.3 at page 7 in Owad and Tsvietkova).
T(n, n-2) = A006331(n-1).
EXAMPLE
The triangle begins
1;
1, 0;
2, 2, 0;
12, 10, 3, 0;
82, 82, 28, 4, 0;
646, 738, 315, 60, 5, 0;
...
MATHEMATICA
bigO[k_, s_]:=Binomial[2s-k-1, k]CatalanNumber[s-k]^2; T[n_, k_]:=Sum[(-1)^(m+k)Binomial[m, k]bigO[m, n], {m, k, n}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]]
CROSSREFS
Cf. A000007 (k = n), A000027 (k = n - 1), A000108, A001246 (row sums), A006331, A007318, A052553.
Sequence in context: A364240 A117270 A244137 * A181389 A369072 A091466
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 18 2022
STATUS
approved