OFFSET
1,5
COMMENTS
Equivalently, the number of "culminating paths", that is, 0-1 sequences of length n-2 such that every prefix has at least as many 1's as 0's, and the difference between the number of 1's and 0's in a prefix is maximal when the prefix is the entire word. - Mireille Bousquet-Mélou, Apr 07 2015
Also, coefficients of the power series for the capacitance of a sphere-plane system, expanded in terms of R/2d where R is the radius of the sphere and d is the distance between the sphere center and the infinite plane. - Scott Crittenden, Dec 15 2017
REFERENCES
L. Page and N. I. Adams, Principles of Electricity, D. Van Nostrand Co., 1931, p. 105.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bousquet-Mélou and Y. Ponty, Culminating paths, Discrete Math and Theoretical Computer Science, Vol. 10, No. 2 (2008) 125-152.
B. Hackl, C. Heuberger, H. Prodinger, and S. Wagner, Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum, arXiv preprint arXiv:1503.08790 [math.CO], 2015.
Y. Zhao, Constructing MSTD sets using bidirectional ballot sequences, arXiv:0908.4442 [math.CO], 2009.
Y. Zhao, Constructing MSTD sets using bidirectional ballot sequences, Journal of Number Theory, Volume 130, Issue 5, May 2010, Pages 1212-1220.
FORMULA
G.f.: Sum_k t^{k+2}/F_{k-1} where F_k is the sequence of Fibonacci polynomials, given by F_0=F_1=1 and F_k=F_{k-1}-t^2 F_{k-2}. - Mireille Bousquet-Mélou, Apr 07 2015
G.f.: Sum_{k>0} (2*x)^k*p/((1+p)^k-(1-p)^k), with p=sqrt(1-4*x^2). - Benedict W. J. Irwin, Sep 15 2016
EXAMPLE
For n=6, the three sequences are 111111, 110111, 111011.
The corresponding culminating paths are 1111, 1011, 1101.
MATHEMATICA
Num=30; p=Sqrt[1-4x^2]; Rest[CoefficientList[Series[Sum[(2^n p x^n)/ ((1+p)^n-(1-p)^n), {n, 1, Num}], {x, 0, Num}], x]] (* Benedict W. J. Irwin, Sep 15 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Yufei Zhao (yufei.zhao(AT)gmail.com), Nov 05 2009, Nov 11 2009
STATUS
approved