A265180 -id:A265180 - cp's OEIS frontend

A259885 a(n) = max{T(n,k), k=1..n}, where T(n,k) is the number of Dyck paths of length 2n and height k (1<=k<=n).

1, 1, 3, 7, 18, 57, 169, 484, 1684, 5661, 18579, 59917, 214058, 760487, 2665884, 9246276, 31945379, 117939506, 431530926, 1567159901, 5655480303, 20299352107, 74300429926, 278279597781, 1037075511926, 3848154018734, 14224439297732, 52402156308977

Offset: 1

Gheorghe Coserea, Jul 07 2015

			For n=4, a(4)=7 because T(4,1)=1, T(4,2)=7, T(4,3)=5, T(4,4)=1.

Gheorghe Coserea, Table of n, a(n) for n = 1..1535

Cf. A080936, A259899 (position of maximum), A265180.

a(n) ~ 4*K/sqrt(Pi) * 4^n/n^2, where K = 0.2675... (see A265180). - Gheorghe Coserea, Dec 05 2015

A265179 Decimal expansion of the position of the maximum of the function f(x) = xSum_{n>=1}n^2(2n^2x^2-3)exp(-n^2x^2).

Original entry on oeis.org

1, 6, 3, 7, 0, 6, 1, 3, 4, 8, 7, 3, 7, 0, 1, 7, 0, 3, 8, 9, 0, 7, 3, 5, 5, 5, 8, 2, 2, 8, 2, 9, 3, 9, 6, 0, 9, 7, 6, 2, 9, 7, 8, 9, 0, 2, 4, 4, 9, 7, 5, 4, 9, 3, 3, 7, 3, 3, 2, 8, 0, 7, 7, 1, 3, 8, 1, 3, 2, 7, 8, 4, 7, 4, 5, 9, 3, 9, 4, 9, 5, 6, 6, 8, 1, 3, 1

Offset: 1

Gheorghe Coserea, Dec 03 2015

			1.63706134...

Gheorghe Coserea, Table of n, a(n) for n = 1..535
Philippe Flajolet, Zhicheng Gao, Andrew M. Odlyzko, Bruce Richmond. The Distribution of heights of binary trees and other simple trees, [Research Report] RR-1749, INRIA. 1992.
Philippe Flajolet, Andrew M. Odlyzko. The average height of binary trees and other simple trees, Research Report RR-0056, INRIA, 1981.

Cf. A259899, A265180 (value of maximum).

Equals lim A259899(n)/sqrt(n).

cp's OEIS Frontend

A259885 a(n) = max{T(n,k), k=1..n}, where T(n,k) is the number of Dyck paths of length 2n and height k (1<=k<=n).

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A265179 Decimal expansion of the position of the maximum of the function f(x) = xSum_{n>=1}n^2(2n^2x^2-3)exp(-n^2x^2).

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cp's OEIS Frontend

A259885 a(n) = max{T(n,k), k=1..n}, where T(n,k) is the number of Dyck paths of length 2n and height k (1<=k<=n).

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A265179 Decimal expansion of the position of the maximum of the function f(x) = x*Sum_{n>=1}n^2*(2*n^2*x^2-3)*exp(-n^2*x^2).

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A265179 Decimal expansion of the position of the maximum of the function f(x) = xSum_{n>=1}n^2(2n^2x^2-3)exp(-n^2x^2).