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A344191
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a(n) = Catalan(n) * (n^2 + 2) / (n + 2).
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7
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1, 1, 3, 11, 42, 162, 627, 2431, 9438, 36686, 142766, 556206, 2169268, 8469060, 33096195, 129454695, 506793270, 1985612310, 7785510810, 30548406570, 119944382220, 471241577820, 1852521913710, 7286586193926, 28675561428972, 112905199767052, 444752335104252
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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)).
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) ~ (2^(2*n - 3)*(8*n - 25)) / (sqrt(Pi)*n^(3/2)). (End)
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EXAMPLE
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All 3 intervals in the poset of cardinality 2 are linear. All 11 intervals in the poset of cardinality 5 are linear.
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MAPLE
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a := n -> `if`(n = 0, 1, a(n-1)*(2*(2*n-1)*(n^2+2))/((n+2)*(n^2-2*n+3))):
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MATHEMATICA
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a[n_] := CatalanNumber[n] (n^2 + 2) / (n + 2);
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PROG
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(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)
(PARI) a(n) = (binomial(2*n, n)/(n+1))*((n^2 + 2)/(n + 2)); \\ Michel Marcus, May 11 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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