Search: id:A002457
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%I A002457 M4198 N1752
%S A002457 1,6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,
%T A002457 67603900,280816200,1163381400,4808643120,19835652870,81676217700,
%U A002457 335780006100,1378465288200,5651707681620,23145088600920,94684453367400
%N A002457 (2n+1)!/n!^2.
%C A002457 Expected number of matches remaining in Banach's modified matchbox problem 
               (counted when last match is drawn from one of the two boxes), multiplied 
               by 4^(n-1). - Michael Steyer (msteyer(AT)osram.de), Apr 13 2001
%C A002457 Sum_{n>=0} 1/a(n) = 2*Pi/3^(3/2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 07 2009]
%C A002457 Hankel transform is (-1)^n*A014480(n). [From Paul Barry (pbarry(AT)wit.ie), 
               Apr 26 2009]
%C A002457 Convolved with A000108: (1, 1, 1, 5, 14, 42,...) = A000531: (1, 7, 38, 
               187, 874,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 
               14 2009]
%C A002457 Convolution of A000302 and A000984 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               May 18 2009]
%C A002457 1.)Number of permutations of three distinct letters (ABC) 0 to n times 
               >>("-". ABC, AABBCC, AAABBBCCC, etc...)and one after the other to 
               resemble motif: AAA (3-0), AAAAAB (5-1), AAAAAAABB (7-2), AAAAAAAAABBB 
               (9-3), AAAAAAAAAAABBBB (11-4), AAAAAAAAAAAAABBBBB (13-5), etc...,
               >>1,6,30,140,630,2772,12012,etc. 1(one) fixed points. Example:motif: 
               AAA (or BBB, or CCC) CAB CBA ACB BCA ABC BAC 6 * 1 fixed point etc... 
               2.) A005408*A000984 example: 7*20 =140 9*70 =630 11*252 =2772 13*924 
               =12012 15*3432 =51480 etc... [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 15 2009]
%D A002457 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 159.
%D A002457 R. Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117.
%D A002457 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25; p. 
               168, #30.
%D A002457 W. Feller, An Introduction to Probability Theory and Its Applications, 
               Vol. I.
%D A002457 C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 449.
%D A002457 M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM 
               Review, SIAM, 1990; see pp. 127-129.
%D A002457 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, 
               p. 514.
%D A002457 A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", 
               Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, 
               Gordon and Breach Science Publishers, 1986-1992.
%D A002457 H. E. Salzer, Coefficients for numerical differentiation with central 
               differences, J. Math. Phys., 22 (1943), 115-135.
%D A002457 J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, 
               Paris, 1933, p. 92.
%D A002457 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002457 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002457 T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, 
               Vol. 2, Engelmann, Leipzig, 1880, p. 21.
%H A002457 T. D. Noe, <a href="b002457.txt">Table of n, a(n) for n=0..200</a>
%H A002457 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CentralBetaFunction.html">Central Beta Function</a>
%H A002457 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 A002457 G.f.: (1-4x)^(-3/2). a(n-1)=binomial(2n, n)*n/2 = binomial(2n-1, n)*n.
%F A002457 a(n-1)=4^(n-1)*sum(binomial(n-1+i, i)*(n-i)/2^(n-1+i), i=0..n-1).
%F A002457 a(n) ~ 2*pi^(-1/2)*n^(1/2)*2^(2*n)*{1 + 3/8*n^-1 + ...} - Joe Keane (jgk(AT)jgk.org), 
               Nov 21 2001
%F A002457 (2*n+2)!/(2*n!*(n+1)!) = (n+n+1)!/(n!*n!) = 1/beta(n+1, n+1) in A061928.
%F A002457 Sum(i * binomial(n, i)^2, i=0.. n) = n*binomial(2*n, n)/2 - Yong Kong 
               (ykong(AT)curagen.com), Dec 26 2000
%F A002457 a(n) ~ 2*pi^(-1/2)*n^(1/2)*2^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 
               07 2002
%F A002457 a(n) = 1/Integral_{x=0..1} x^n (1-x)^n dx. - Fred W. Helenius (fredh(AT)ix.netcom.com), 
               Jun 10 2003
%F A002457 E.g.f.: exp(2*x)*((1+4*x)*BesselI(0, 2*x)+4*x*BesselI(1, 2*x)). - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Sep 22 2003
%F A002457 a(n)=sum(i+j+k=n, binomial(2i, i)binomial(2j, j)binomial(2k, k)) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Nov 09 2003
%F A002457 Equals (2*n+1)*A000984(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 12 2006
%F A002457 a(n-1)=Sum_{k, 0<=k<=n}A039599(n,k)*A000217(k), for n>=1 . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007
%F A002457 Sum of (n+1)-th row terms of triangle A132818. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 02 2007
%F A002457 a(n)=sum{k=0..n, C(2k,k)*4^(n-k)}. [From Paul Barry (pbarry(AT)wit.ie), 
               Apr 26 2009]
%p A002457 A002457 := n-> (n+1)*binomial(2*(n+1), (n+1))/2;
%p A002457 a:=n->sum(abs(binomial(-n,-2*n)),j=1..n): seq(a(n),n=1..23); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
%p A002457 a:=n->abs(sum((binomial(-n,n-2)),j=2..n)): seq(a(n),n=2..24); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
%p A002457 restart:a:= proc(n) option remember; if n=0 then 1 else add((binomial 
               (2*n,n))/2, j=0..n-1) fi end: seq (a(n), n=1..23);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2009]
%t A002457 a[n_]:=(2*n+1)!/n!^2; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Dec 13 2008]
%o A002457 (PARI) a(n)=if(n<0,0,(2*n+1)!/n!^2)
%Y A002457 Cf. A033876. Also a(n)=f(n, n-3) where f is given in A034261.
%Y A002457 Denominator of central elements of Leibniz's Harmonic Triangle A003506.
%Y A002457 Cf. A000531 (Banach's original match problem). Equals A002011/4.
%Y A002457 a(n) = A005430(n+1)/2 = A002011(n)/4.
%Y A002457 Cf. A000984, A001803, A132818.
%Y A002457 A000531 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 14 2009]
%Y A002457 Sequence in context: A125316 A092439 A082149 this_sequence A137400 A026749 
               A003279
%Y A002457 Adjacent sequences: A002454 A002455 A002456 this_sequence A002458 A002459 
               A002460
%K A002457 nonn,easy,nice
%O A002457 0,2
%A A002457 N. J. A. Sloane (njas(AT)research.att.com).

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