%I #45 Feb 20 2025 08:37:26
%S 1,6,5,28,9,22,13,120,17,38,21,92,25,54,29,496,33,70,37,156,41,86,45,
%T 376,49,102,53,220,57,118,61,2016,65,134,69,284,73,150,77,632,81,166,
%U 85,348,89,182,93,1520,97,198,101,412,105,214,109,888,113,230,117
%N Coefficients related to an asymptotic expansion of the logarithm of the central binomial.
%C An asymptotic expansion of the logarithm of the central binomial (A000984) for n>0 is given by log(binomial(2*n,n)) ~ (n*log(16)-log(Pi)-log(n) + sum_{k>=1}((-4)^(-k)*A002425(k)/a(k)*n^(1-2*k)))/2.
%C An asymptotic expansion of the logarithm of the swinging factorial (A056040) for n>1 is given by log(swing(n)) ~ (n*log(4)-log(Pi)-(-1)^n*(log(n/2) - (1/2)*sum_{k>=1}((-1)^k*A002425(k)/a(k)*n^(1-2*k))))/2.
%H Peter Luschny, <a href="/A219931/b219931.txt">Table of n, a(n) for n = 1..300</a>
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%F a(2^p*n - 2^(p-1)) = 2^(p-1)*(2^p-1) + 4^p*(n-1) for p >= 1. - _Johannes W. Meijer_, Dec 09 2012
%F a(n) = denominator(2*E(2*n+1,1)/(2*n+1)) where E(n,x) is the Euler polynomial. - _Peter Luschny_, Apr 03 2014
%F a(n) = (2*n-1)*A006519(n). - _Flávio V. Fernandes_, Feb 19 2025
%e log(binomial(2*n,n)) = n*log(4) - (log(n)+log(Pi))/2 - 1/(8*a(1)*n) + 1/(32*a(2)*n^3) - 1/(128*a(3)*n^5) + 17/(512*a(4)*n^7) - 31/(2048*a(5)*n^9) + 691/(8192*a(6)*n^11) + O(1/n^13).
%e log(swing(n)) = n*log(2) - (1/2)*log(Pi) - (1/4)*(-1)^n*(2*log(n/2) + 1/(a(1)*n) - 1/(a(2)*n^3) + 1/(a(3)*n^5) - 17/(a(4)*n^7) + 31/(a(5)*n^9) - 691/(a(6)*n^11)) + O(1/n^13).
%p Coeff_list := proc(len) local n;
%p asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
%p subs(n=1/n,simplify(convert(%,polynom)));
%p [seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
%p A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);
%p # second Maple program:
%p A006516 := n -> 2^(n-1)*(2^n-1): A029837 := n -> ceil(simplify(log[2](n))): nmax:=59: for n from 1 to nmax do for p from 1 to A029837(nmax) do a(2^p*n - 2^(p-1)) := A006516(p) + 4^p*(n-1) od: od: seq(a(n), n=1..nmax); # _Johannes W. Meijer_, Dec 09 2012
%t max = 60; s = Normal[Series[Log[x/2]/2+LogGamma[x/2+1/2]-LogGamma[x/2+1], {x, Infinity, 2*max}]] /. x -> 1/x; a[n_] := Denominator[4*Coefficient[s, x^(2*n-1), 1]]; Table[a[n], {n, 1, max}] (* _Jean-François Alcover_, Feb 17 2014 *)
%t a[n_] := Denominator[2*EulerE[2*n-1, 1]/(2*n-1)]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Apr 04 2014, after _Peter Luschny_ *)
%o (Sage)
%o def A219931_list(len): # After Johannes W. Meijer
%o z = s = 1; a = {}; p = len + 1
%o while p > 0:
%o p >>= 1; n = 0; i = z; z = z*2;
%o s = s*4; u = (s-z)/2
%o while i <= len:
%o a[i] = u + s*n
%o i += z; n += 1
%o return [a[i] for i in (1..len)]
%o A219931_list(59) # _Peter Luschny_, Dec 09 2012
%o (PARI) a(n) = (2*n-1)*2^valuation(n,2) \\ _Andrew Howroyd_, Feb 19 2025
%Y Cf. A006519, A118413.
%K nonn,easy,changed
%O 1,2
%A _Peter Luschny_, Dec 01 2012