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A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.

%I A219931 #56 Jun 17 2025 08:14:29
%S A219931 1,6,5,28,9,22,13,120,17,38,21,92,25,54,29,496,33,70,37,156,41,86,45,
%T A219931 376,49,102,53,220,57,118,61,2016,65,134,69,284,73,150,77,632,81,166,
%U A219931 85,348,89,182,93,1520,97,198,101,412,105,214,109,888,113,230,117
%N A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial.
%C A219931 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 A219931 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 A219931 Peter Luschny, <a href="/A219931/b219931.txt">Table of n, a(n) for n = 1..300</a>
%H A219931 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%F A219931 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 A219931 Warning: a(n) != (2*n-1)*2^valuation(n, 2). This was mistakenly assumed several times in the past, see A385054. - _Peter Luschny_, Jun 17 2025
%e A219931 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 A219931 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 A219931 Coeff_list := proc(len) local n;
%p A219931 asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1),n,2*len+3);
%p A219931 subs(n=1/n,simplify(convert(%,polynom)));
%p A219931 [seq(4*coeff(unapply(%,n)(n),n,2*k+1),k=0..len-1)] end:
%p A219931 A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);
%t A219931 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 A219931 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_ *)
%Y A219931 Cf. A006519, A118413, A385054.
%K A219931 nonn,easy
%O A219931 1,2
%A A219931 _Peter Luschny_, Dec 01 2012
%E A219931 Edited and incorrect entries removed by _Georg Fischer_ and _Peter Luschny_, Jun 16 2025