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A219931 Coefficients related to an asymptotic expansion of the logarithm of the central binomial. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Dec 01 2012
STATUS
approved

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Last modified July 23 11:07 EDT 2024. Contains 374549 sequences. (Running on oeis4.)