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%I #63 Dec 18 2022 07:59:23
%S 0,1,1,9,5,175,189,1617,429,57915,60775,508079,264537,8788507,9100525,
%T 75218625,9694845,5109183315,5250613995,43106892675,22090789875,
%U 723694276305,740104577355,6049284520695,1543768261425,201547523019375
%N 0 followed by the numerators of the reduced (A001803(n) + A001790(n)) / (2*A046161(n)).
%C We write the fractions a(n)/b(n) and higher order differences as a matrix:
%C 0, 1, 1, 9/8, 5/4,...
%C 1, 0, 1/8, 1/8, 15/128,... = (A001790(n)/A046161(n) +Lorbeta(n)) /2
%C -1, 1/8, 0, -1/128, -1/128,... = (Lorbeta(n+1) +A161200(n+1)/A046161(n+1)) / 2
%C 9/8, -1/8, -1/128, 0, 1/1024,...
%C -5/4, 15/128, 1/128, 1/1024, 0,...
%C Here, Lorbeta(0)=1 and Lorbeta(n) = -A098597(n-1)/A046161(n) for n>0 is the inverse of the Lorentz factor.
%C The first line with numerators a(n) and denominators b(n) is 0, 1, 1, 9/8, 5/4, 175/128, 189/128, 1617/1024, 429/256, 57915/32768, 60775/32768,... It is an autosequence: Its inverse binomial transform is the signed sequence.
%C a(n+1)/(2*n-1)= 1, 3, 1, 25, 21, 147, 33, 3861, 3575, 26741,... .
%C a(n+1)/A146535(n) = 9, 5, 35, 27, 539, 39, 4455,... .
%C A001790(n)/A046161(n) yields the coefficients of the Lorentz factor (or Lorentz gamma factor). With b for beta and g for gamma:
%C g = (1-b^2)^-1 = 1 + (b^2)/2 + 3*(b^4)/8 + 5*(b^6)/16 + ... .
%C b = (1-g^-2)^-1 = 1 - (g^-2)/2 - (g^-4)/8 - (g^-6)/16 - ... .
%C Are the denominators of the first subdiagonal 1, 1/8, -1/128, 1/1024,... A061549(n) ?
%C a(n+1)/(A000108(n)*b(n)) = 1, 1, 9/16, 1/4, 25/256, 9/256, 49/4096, 1/256, 81/65536, 25/65536, 121/1048576,... = A191871(n+1)/ A084623(n+1)^2 ?
%H Vincenzo Librandi, <a href="/A206771/b206771.txt">Table of n, a(n) for n = 0..1000</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lorentz factor">Lorentz Factor</a>.
%F a(n) = A000265(n) * A001790(n-1).
%F a(n) = A098597(n-1) * A191871(n). See also A181318
%F a(n) = numerator of n*binomial(2n-2,n-1)/4^(n-1). - _Nathaniel Johnston_, Dec 16 2022
%e From the first formula: a(1)=1*1, a(2)=1*1, a(3)=3*3, a(4)=1*5, a(5)=5*35, a(6)=3*63.
%p A206771 := proc(n)
%p A001790(n)+A001803(n) ;
%p %/2/A046161(n) ;
%p numer(%) ;
%p end proc: # _R. J. Mathar_, Jan 18 2013
%t max = 25; A001803 = CoefficientList[Series[(1 - x)^(-3/2), {x, 0, max}], x] // Numerator; A001790 = CoefficientList[Series[1/Sqrt[(1 - x)], {x, 0, max}], x] // Numerator; A046161 = Table[Binomial[2n, n]/4^n, {n, 0, max}] // Denominator; a[n_] := (A001803[[n]] + A001790[[n]])/(2*A046161[[n]]) // Numerator; a[0] = 0; Table[a[n], {n, 0, max}]
%t (* or (from 1st formula) : *) Table[ n*Numerator[4^(1-n)*Binomial[2n-2, n-1]]/2^IntegerExponent[n, 2], {n, 0, max}]
%t (* or (from 2nd formula) : *) Table[ Numerator[ CatalanNumber[n-1]/2^(2n-1)]*Numerator[n^2/2^n], {n, 0, max}] (* _Jean-François Alcover_, Jan 31 2013 *)
%o (Magma) /* By definition: */ m:=25; R<x>:=PowerSeriesRing(Rationals(), m); p:=Coefficients(R!(1/(1-x)^(1/2))); q:=Coefficients(R!((1-x)^(-3/2))); A001790:=[Numerator(p[i]): i in [1..m]]; A001803:=[Numerator(q[i]): i in [1..m]]; A046161:=[Denominator(Binomial(2*n,n)/4^n): n in [0..m-1]]; [0] cat [Numerator((A001803[n]+A001790[n])/(2*A046161[n])): n in [1..m]]; // _Bruno Berselli_, Mar 11 2013
%Y Cf. A187791, A000108.
%K nonn,frac
%O 0,4
%A _Paul Curtz_, Jan 10 2013
%E a(11)-a(25) from _Jean-François Alcover_, Jan 13 2013
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