OFFSET
0,2
COMMENTS
Numerator of e(0,n) (see the Maple line).
The generating function of the full fraction is (1-2*x)^(-3/4). - R. J. Mathar, Nov 06 2011
LINKS
V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
FORMULA
Numerator of 2^n*Gamma(n + 3/4)/(Gamma(3/4)*n!). - R. J. Mathar, Nov 06 2011
Numerator of integral_{x>0} 1/(x^4 + 1)^(n+1) / (Pi*sqrt(2)). - Jean-François Alcover, Apr 29 2013
From Petros Hadjicostas, May 23 2020: (Start)
If fr(n) = A067002(n)/A046161(n), then fr(n) = P_n(0), where P_n(x) is the Boros-Moll polynomial mentioned in A223549 and A223550 (and whose coefficients are the numbers e(l,n) = A067001(n,n-l)/2^(2*n) that are mentioned in the Maple line below with l = 0..n).
Recurrence for fr(n): 4*n*(n - 1)*fr(n) = 8*(2*n - 1)*(n - 1)*fr(n-1) - (16*(n-1)^2 - 1)*fr(n-2) for n >= 2 with fr(0) = 1 and fr(1) = 3/2. (End)
EXAMPLE
MAPLE
e := proc(l, m) local k; add(2^(k-2*m)*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m); end;
MATHEMATICA
Numerator[Table[Sum[2^(k-2n) Binomial[2n-2k, n-k]Binomial[n+k, n], {k, 0, n}], {n, 0, 30}]] (* Harvey P. Dale, Oct 19 2012 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Feb 16 2002
STATUS
approved