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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..54. 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 A091466 A134085 Adjacent sequences: A354124 A354125 A354126 * A354128 A354129 A354130 KEYWORD nonn,tabl AUTHOR Stefano Spezia, May 18 2022 STATUS approved

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Last modified November 29 20:23 EST 2023. Contains 367447 sequences. (Running on oeis4.)