A121686 -id:A121686 - cp's OEIS frontend

A344191 a(n) = Catalan(n) * (n^2 + 2) / (n + 2).

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

Offset: 0

F. Chapoton, May 11 2021

nonn

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.

			All 3 intervals in the poset of cardinality 2 are linear. All 11 intervals in the poset of cardinality 5 are linear.

Michael De Vlieger, Table of n, a(n) for n = 0..1664
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
Clément Chenevière, Enumerative study of intervals in lattices of Tamari type, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 150.
J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003.

Cf. A000108, A002694, A127632, A344136, A121686.

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

a[n_] := CatalanNumber[n] (n^2 + 2) / (n + 2);
Table[a[n], { n, 0, 23}] (* Peter Luschny, May 11 2021 *)

a(n) = (binomial(2*n,n)/(n+1))*((n^2 + 2)/(n + 2)); \\ Michel Marcus, May 11 2021

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))

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

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

A121685 Triangle read by rows: T(n,k) is the number of binary trees having n edges and k branches (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.

Original entry on oeis.org

2, 4, 1, 8, 4, 2, 16, 12, 12, 2, 32, 32, 48, 16, 4, 64, 80, 160, 80, 40, 5, 128, 192, 480, 320, 240, 60, 10, 256, 448, 1344, 1120, 1120, 420, 140, 14, 512, 1024, 3584, 3584, 4480, 2240, 1120, 224, 28, 1024, 2304, 9216, 10752, 16128, 10080, 6720, 2016, 504, 42

Offset: 1

Emeric Deutsch, Aug 15 2006

The row sums are the Catalan numbers (A000108). T(n,1)=2^n = A000079(n). T(n,n)=A089408(n+1). Sum(k*T(n,k),k=1..n)=A121686(n).

			Triangle starts:
2;
4,1;
8,4,2;
16,12,12,2;
32,32,48,16,4;

Cf. A000108, A000079, A089408, A121686.

c:=n->binomial(2*n,n)/(n+1): T:=proc(n,k) if k mod 2 = 0 then c(k/2)*binomial(n-1,k-1)*2^(n-k) else c((k-1)/2)*binomial(n-1,k-1)*2^(n-k+1) fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

T(n,k)=2^(n-k)*c(k/2)*binomial(n-1,k-1) if k is even and 2^(n-k+1)*c((k-1)/2)*binomial(n-1,k-1) if k is odd, where c(m)=binomial(2m,m)/(m+1) are the Catalan numbers (A000108). G.f.=(1-2z+2tz)(1-2z-sqrt[(1-2z)^2-4t^2*z^2])/(2t^2*z^2) - 1.

cp's OEIS Frontend

A344191 a(n) = Catalan(n) * (n^2 + 2) / (n + 2).

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