OFFSET
0,4
COMMENTS
Number of "reduced 3-historic trees" with n internal vertices: trees with vertex labels from 1 to n that increase from the root to the leaves, and where vertices at an even distance from the root (including the root itself) can only have at most one child, while vertices at an odd distance from the root can have a left child, a right child, both, or no children (see Burghart-Wagner for details). - Stephan Wagner, Feb 06 2026
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..518 (first 200 terms from Alois P. Heinz)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
O. Bodini, M. Dien, X. Fontaine, A. Genitrini, and H. K. Hwang, Increasing Diamonds, in LATIN 2016: 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings Pages pp 207-219 2016 Lecture Notes in Computer Science Series Volume 9644.
F. Burghart and S. Wagner, A Bijection for the Evolution of B-Trees, in 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 10:1-10:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2024).
FORMULA
a(n) ~ c * d^n * n! * n, where d = 0.42089835222875301896706732846764190595145230471243866202153775712470703269... is the root of the equation WeierstrassP(1/d, 0, -1/108) = 1/6 and c = 6d^2 = 1.06293253745327664869312823202016275205862332741406172188742740834633... - Vaclav Kotesovec, Sep 06 2014, updated Feb 06 2026
E.g.f.: 6^(1/3) * WeierstrassP((x+c)/6^(1/3), 0, -1/3), where c = 9.1898572290187191497581591181140131456801040793456712149069964791654... is the root of the equation WeierstrassP(c/6^(1/3), 0, -1/3) = 6^(-1/3). - Vaclav Kotesovec, Jun 14 2015
E.g.f. A(x) satisfies: A(x) = 1 + x + Integral(Integral A(x)^2 dx) dx. - Ilya Gutkovskiy, Jul 04 2020
MAPLE
a:= proc(n) option remember;
`if`(n<2, 1, add(a(i)*a(n-2-i) *binomial(n-2, i), i=0..n-2))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jun 22 2012
MATHEMATICA
a[n_] := a[n] = If[n < 2, 1, Sum[a[i] * a[n - 2 - i] * Binomial[n - 2, i], {i, 0, n - 2}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
Table[SeriesCoefficient[1 + (18 (WeierstrassP[x, {0, -1/108}] - WeierstrassPPrime[x, {0, -1/108}]))/(6 WeierstrassP[x, {0, -1/108}] - 1)^2, {x, 0, k}] k!, {k, 0, 30}] (* Jan Mangaldan, Nov 27 2020 *)
CROSSREFS
KEYWORD
nonn,nice,eigen
AUTHOR
STATUS
approved