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 A344191 a(n) = Catalan(n) * (n^2 + 2) / (n + 2). 6
 1, 1, 3, 11, 42, 162, 627, 2431, 9438, 36686, 142766, 556206, 2169268, 8469060, 33096195, 129454695, 506793270, 1985612310, 7785510810, 30548406570, 119944382220, 471241577820, 1852521913710, 7286586193926 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: These are the number of linear intervals in Pallo's comb posets. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 36686 for n = 9. LINKS J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003. FORMULA a(n) = Catalan(n) + (1/(n + 2))*Sum_{k=2..n}((2^(n - k)*(n - k + 4)/(k - 2)!)* Product_{i=2..k-1}(n + i)). From Peter Luschny, May 11 2021: (Start) a(n) = [x^n] ((2*x + sqrt(1 - 4*x) - 1)*(3*x - 1))/(2*sqrt(1 - 4*x)*x^2). a(n) = n! * [x^n] exp(2*x)*(BesselI(0, 2*x) - BesselI(1, 2*x) + BesselI(2, 2*x)). a(n) = a(n-1)*(2*(2*n - 1)*(n^2 + 2))/((n + 2)*(n^2 - 2*n + 3)) for n >= 1. a(n) = Catalan(n) + binomial(2*n, n-2) = A000108(n) + A002694(n). a(n) ~ (2^(2*n - 3)*(8*n - 25)) / (sqrt(Pi)*n^(3/2)). (End) a(n) = A121686(n) / 2. - Hugo Pfoertner, May 11 2021 EXAMPLE All 3 intervals in the poset of cardinality 2 are linear. All 11 intervals in the poset of cardinality 5 are linear. MAPLE a := n -> `if`(n = 0, 1, a(n-1)*(2*(2*n-1)*(n^2+2))/((n+2)*(n^2-2*n+3))): seq(a(n), n = 0..19); # Peter Luschny, May 11 2021 MATHEMATICA a[n_] := CatalanNumber[n] (n^2 + 2) / (n + 2); Table[a[n], { n, 0, 23}] (* Peter Luschny, May 11 2021 *) PROG (Sage) def a(n):     return catalan_number(n)+sum(2**(n-k)/factorial(k-2)*(n-k+4)/(n+2)*prod(n+i for i in range(2, k)) for k in range(2, n+1)) (Sage) def a(n): return catalan_number(n) + binomial(2*n, n-2) print([a(n) for n in range(24)]) # Peter Luschny, May 11 2021 (PARI) a(n) = (binomial(2*n, n)/(n+1))*((n^2 + 2)/(n + 2)); \\ Michel Marcus, May 11 2021 CROSSREFS Cf. A000108, A002694, A127632, A344136, A121686. Sequence in context: A301483 A059716 A122368 * A032443 A180907 A143464 Adjacent sequences:  A344188 A344189 A344190 * A344192 A344193 A344194 KEYWORD nonn AUTHOR F. Chapoton, May 11 2021 STATUS approved

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Last modified September 24 09:10 EDT 2021. Contains 347630 sequences. (Running on oeis4.)