%I #97 Jun 14 2018 14:53:21
%S 1,7,38,187,874,3958,17548,76627,330818,1415650,6015316,25413342,
%T 106853668,447472972,1867450648,7770342787,32248174258,133530264682,
%U 551793690628,2276098026922,9373521044908,38546133661492
%N From area of cyclic polygon of 2n + 1 sides.
%C Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - _Michael Steyer_, Apr 13 2001
%C A conjectured definition: Let 0 < a_1 < a_2 <...<a_{2n} < 1. Then how many ways are there in which one can add or subtract all the a_i to get an odd number. For example, take n = 2. Then the options are a_1+a_2+a_3+a_4 = 1 or 3; one can change the sign of any of the a_i's and get 1; or -a_1-a_2+a_3+a_4 = 1. That's a total of 7, which is the 2nd number of this sequence. One of the definitions of the sequence (which was how I came across it) is the degree of the equation giving the area of a cyclic polygon in terms of the sides. I conjectured that for any set of side lengths there is a unique way of fitting them together for any possible winding number and any possible subset of sides which go round the circle in a retrograde manner. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
%C a(n) = total weight of upsteps in all Dyck n-paths (A000108) when each upstep is weighted with its position in the path. For example, the Dyck path UDUUDUDD has upsteps in positions 1,3,4,6 and contributes 1+3+4+6=14 to the weight for Dyck 4-paths. The summand (n-k)*binomial(2*n+1, k) in the Maple formula below is the total weight of upsteps terminating at height n-k, 0<=k<=n-1. - _David Callan_, Dec 29 2006
%C Catalan transform of binomial transform of squares. - _Philippe Deléham_, Oct 31 2008
%C a(n) is also the number of walks of length 2n in the quarter plane starting and ending at the origin using steps {(1,1),(1,0),(-1,0), (-1,-1)} (which appear in Gessel's conjecture) in which the steps (1,0) and (-1,0) appear exactly once each. - _Arvind Ayyer_, Mar 02 2009
%C Equals the Catalan sequence, A000108, convolved with A002457: (1, 6, 30, 140, ...). - _Gary W. Adamson_, May 14 2009
%C Total number of occurrences of the pattern 213 (or 132) in all skew-indecomposable (n+2)-permutations avoiding the pattern 123. For example, a(1) = 1, since there is one occurrence of the pattern 213 in the set {213, 132}. - _Cheyne Homberger_, Mar 13 2013
%D W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.
%H T. D. Noe, <a href="/A000531/b000531.txt">Table of n, a(n) for n = 1..100</a>
%H Arvind Ayyer, <a href="http://arxiv.org/abs/0902.2329">Towards a human proof of Gessel's conjecture</a>, arXiv:0902.2329 [math.CO], 2009, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ayyer/ayyer7.html">JIS 12 (2009) 09.4.2</a>
%H Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, <a href="https://arxiv.org/abs/1703.00323">On the self-convolution of generalized Fibonacci numbers</a>, arXiv:1703.00323 [math.CO], 2017.
%H F. Bowman, <a href="http://www.jstor.org/stable/3608200">Cyclic pentagons</a>, Math. Gaz. 36, (1952). 244-250. MR0051523.
%H A. Burstein and S. Elizalde, <a href="http://arxiv.org/abs/1305.3177">Total occurrence statistics on restricted permutations</a>, arXiv preprint arXiv:1305.3177 [math.CO], 2013.
%H C. Homberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p43">Expected patterns in permutation classes</a>, Electronic Journal of Combinatorics, 19(3) (2012), P43.
%H Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%H Yasuhiko Kamiyama, <a href="https://arxiv.org/abs/1803.05559">The Euler characteristic of the regular spherical polygon spaces</a>, arXiv:1803.05559 [math.GT], 2018.
%H A. F. Moebius, <a href="http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?sid=a6ca480904b4c265d485a1da0bbff12a&idno=aax2934.0001.001&c=umhistmath&cc=umhistmath&seq=430&view=image">Über die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fläche des Vielecks gefunden werden</a>, Gesammelte Werke, vol. 1., pp. 407-438.
%H D. P. Robbins, <a href="http://www.jstor.org/stable/2974766">Areas of polygons inscribed in a circle</a>, Amer. Math. Monthly, 102 (1995), 523-530.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclicPolygon.html">Cyclic Polygon</a>
%H Y. Q. Zhao, <a href="http://mathstat.carleton.ca/~zhao/TEACHING/70.265/random-v/random-v.html">Introduction to Probability with Applications</a>
%F a(n) = ((2n+1)!/((n!)^2)-4^n)/2. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
%F na(n) = (8n-2)a(n-1) - (16n-8)a(n-2), n>1. - _Michael Somos_, Apr 18 2003
%F E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x) + 4*x*exp(2*x)*BesselI(1, 2*x) - exp(4*x)). - _Vladeta Jovovic_, Sep 22 2003
%F a(n-1) = 4^n*sum_{k=0..n} binomial(2*k+1, k)*4^(-k) = (2*n+1)*(2*n+3)*C(n) - 2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108. - _Wolfdieter Lang_
%F a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - _Philippe Deléham_, Jun 10 2007
%F a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - _Philippe Deléham_, Oct 31 2008
%F (Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - _L. Edson Jeffery_, Jan 21 2012
%p f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;
%t a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* _Jean-François Alcover_, Dec 07 2011, after Pari *)
%o (PARI) a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
%Y Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431.
%K nonn,easy,nice
%O 1,2
%A _Simon Plouffe_
%E Moebius reference from _Michael Somos_