0,2

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

Sum_{n>=0} 1/a(n) = 2*Pi/3^(3/2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 07 2009]

Hankel transform is (-1)^n*A014480(n). [From Paul Barry (pbarry(AT)wit.ie), Apr 26 2009]

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]

Convolution of A000302 and A000984 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 18 2009]

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]

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 159.

R. Chapman, Moments of Dyck paths, Discrete Math., 204 (1999), 113-117.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25; p. 168, #30.

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 449.

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.

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.

H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.

J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21.

T. D. Noe, Table of n, a(n) for n=0..200

Eric Weisstein's World of Mathematics, Central Beta Function

Y. Q. Zhao, Introduction to Probability with Applications

G.f.: (1-4x)^(-3/2). a(n-1)=binomial(2n, n)*n/2 = binomial(2n-1, n)*n.

a(n-1)=4^(n-1)*sum(binomial(n-1+i, i)*(n-i)/2^(n-1+i), i=0..n-1).

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

(2*n+2)!/(2*n!*(n+1)!) = (n+n+1)!/(n!*n!) = 1/beta(n+1, n+1) in A061928.

Sum(i * binomial(n, i)^2, i=0.. n) = n*binomial(2*n, n)/2 - Yong Kong (ykong(AT)curagen.com), Dec 26 2000

a(n) ~ 2*pi^(-1/2)*n^(1/2)*2^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 07 2002

a(n) = 1/Integral_{x=0..1} x^n (1-x)^n dx. - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 10 2003

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

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

Equals (2*n+1)*A000984(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 12 2006

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

Sum of (n+1)-th row terms of triangle A132818. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 02 2007

a(n)=sum{k=0..n, C(2k,k)*4^(n-k)}. [From Paul Barry (pbarry(AT)wit.ie), Apr 26 2009]

A002457 := n-> (n+1)*binomial(2*(n+1), (n+1))/2;

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

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

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]

a[n_]:=(2*n+1)!/n!^2; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]

(PARI) a(n)=if(n<0, 0, (2*n+1)!/n!^2)

Cf. A033876. Also a(n)=f(n, n-3) where f is given in A034261.

Denominator of central elements of Leibniz's Harmonic Triangle A003506.

Cf. A000531 (Banach's original match problem). Equals A002011/4.

a(n) = A005430(n+1)/2 = A002011(n)/4.

Cf. A000984, A001803, A132818.

A000531 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 14 2009]

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).