|
%I #75 Mar 24 2022 08:04:49
%S 0,2,10,42,168,660,2574,10010,38896,151164,587860,2288132,8914800,
%T 34767720,135727830,530365050,2074316640,8119857900,31810737420,
%U 124718287980,489325340400,1921133836440,7547311500300,29667795388452,116686713634848
%N a(n) = binomial(2*n, n) mod binomial(2*n-2, n-1).
%C Apart from its root term -1: central terms of the triangle in A051631: a(n) = A051631(2*(n-1), n-1). - _Reinhard Zumkeller_, Nov 13 2011
%C Define an array m(i,j) by m(1,j)=m(j,1)=j*(j+1)/2 for j=0,1,2,3,... and m(i,j) = m(i,j-1) + m(i-1,j+1); the diagonal m(k,k) for k=1,2,3... gives the numbers in this sequence. - _J. M. Bergot_, May 02 2012
%C The central terms of triangle A051631 (including the root term -1) are given by (n-1)*(n+1)*Gamma(2*n+1)/Gamma(n+2)^2 with n >= 0. - _Peter Luschny_, Nov 24 2013
%C Index the sequence from n=0 so that a(0)=1, a(1)=0, a(2)=2, a(3)=10, ... a(n) is the number of walks using steps U=(1,1) and D=(1,-1) from the origin to (2n,0) that rise above and dip below the x axis. a(2) = 2 because we have: DUUD and UDDU. - _Geoffrey Critzer_, Jan 11 2014
%H Reinhard Zumkeller, <a href="/A024483/b024483.txt">Table of n, a(n) for n = 2..1000</a>
%F a(n) = ((n-2)/n)*binomial(2*n-2, n-1) = (n-2)*A000108(n-1). - _Vladeta Jovovic_, Aug 03 2002
%F a(n) = 2*binomial(2n-3, n-3) = 2*A002054(n-2). - _Ralf Stephan_, Jan 15 2004
%F a(n) = A000984(n-1) - 2*A000108(n-1). - _Geoffrey Critzer_, Jan 11 2014
%F a(n) ~ 4^(n-1)/sqrt(Pi*n). - _Ilya Gutkovskiy_, Sep 13 2016
%F D-finite with recurrence n*a(n) +(-7*n+8)*a(n-1) +6*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Apr 27 2020
%F From _Amiram Eldar_, Mar 24 2022: (Start)
%F Sum_{n>=3} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
%F Sum_{n>=3} (-1)^(n+1)/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)
%p seq((n-1)*binomial(2*n, n)/(n+1), n=1..25); # _Zerinvary Lajos_, Feb 28 2007
%t nn=20; d=(1-(1-4x)^(1/2))/(2x); Drop[CoefficientList[Series[1/(1-2x d)-2(d-1), {x,0,nn}],x],1] (* _Geoffrey Critzer_, Jan 11 2014 *)
%t Table[Mod[Binomial[2 n, n], Binomial[2 n - 2, n - 1]], {n, 2, 26}] (* _Michael De Vlieger_, Sep 13 2016 *)
%o (Haskell)
%o a024483 n = a051631 (2*(n-1)) (n-1) -- _Reinhard Zumkeller_, Nov 13 2011
%o (Sage)
%o def a(n): return n*(n-2)*factorial(2*(n-1))/factorial(n)^2
%o [a(n) for n in (2..26)] # _Peter Luschny_, Nov 24 2013
%Y Cf. A000108, A000984, A001622, A051631.
%K nonn,easy
%O 2,2
%A _Clark Kimberling_
%E More terms from _Zerinvary Lajos_, Oct 02 2007
|
