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 A000531 From area of cyclic polygon of 2n + 1 sides. 18
 1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 A conjectured definition: Let 0 < a_1 < a_2 <...1. - Michael Somos, Apr 18 2003 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 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 a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - Philippe Deléham, Jun 10 2007 a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - Philippe Deléham, Oct 31 2008 (Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - L. Edson Jeffery, Jan 21 2012 MAPLE f := proc(n) sum((n-k)*binomial(2*n+1, k), k=0..n-1); end; MATHEMATICA a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Dec 07 2011, after Pari *) PROG (PARI) a(n)=if(n<1, 0, ((2*n+1)!/n!^2-4^n)/2) CROSSREFS Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431. Sequence in context: A249021 A114290 A277912 * A296769 A241524 A291822 Adjacent sequences:  A000528 A000529 A000530 * A000532 A000533 A000534 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Moebius reference from Michael Somos STATUS approved

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Last modified October 22 17:33 EDT 2019. Contains 328319 sequences. (Running on oeis4.)