A219931 - OEIS

A219931

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

1, 6, 5, 28, 9, 22, 13, 120, 17, 38, 21, 92, 25, 54, 29, 496, 33, 70, 37, 156, 41, 86, 45, 376, 49, 102, 53, 220, 57, 118, 61, 2016, 65, 134, 69, 284, 73, 150, 77, 632, 81, 166, 85, 348, 89, 182, 93, 1520, 97, 198, 101, 412, 105, 214, 109, 888, 113, 230, 117

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OFFSET

1,2

COMMENTS

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.

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.

LINKS

Peter Luschny, Table of n, a(n) for n = 1..300

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

FORMULA

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

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

EXAMPLE

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).

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).

MAPLE

Coeff_list := proc(len) local n;

asympt(ln(n/2)/2+lnGAMMA(n/2+1/2)-lnGAMMA(n/2+1), n, 2*len+3);

subs(n=1/n, simplify(convert(%, polynom)));

[seq(4*coeff(unapply(%, n)(n), n, 2*k+1), k=0..len-1)] end:

A219931_list := n -> denom(Coeff_list(n)); A219931_list(59);

# second Maple program:

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

MATHEMATICA

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 *)

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 *)

PROG

(Sage)

def A219931_list(len): # After Johannes W. Meijer

z = s = 1; a = {}; p = len + 1

while p > 0:

p >>= 1; n = 0; i = z; z = z*2;

s = s*4; u = (s-z)/2

while i <= len:

a[i] = u + s*n

i += z; n += 1

return [a[i] for i in (1..len)]