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

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)]
A219931_list(59)  # Peter Luschny, Dec 09 2012

Crossrefs

Cf. A118413.

Sequence in context: A371253 A267743 A298155 * A296192 A070399 A137763

Adjacent sequences: A219928 A219929 A219930 * A219932 A219933 A219934

Keyword

nonn

Author

Peter Luschny, Dec 01 2012

Status

approved