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%I #26 Jul 08 2021 11:20:27
%S 1,3,21,77,1155,4389,33649,129789,4023459,15646785,122044923,
%T 477084699,7474326951,29322359577,230389968105,906200541213,
%U 57090634096419,225004263791769,1775033636579511,7006711723340175,110706045228774765
%N Numerator of Sum_{k=0..n} 2^(k-2*n) * binomial(2*n-2*k,n-k) * binomial(n+k,n).
%C Numerator of e(0,n) (see the Maple line).
%C The generating function of the full fraction is (1-2*x)^(-3/4). - _R. J. Mathar_, Nov 06 2011
%H V. H. Moll. <a href="http://www.ams.org/notices/200203/index.html">The evaluation of integrals: a personal story</a>, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
%F Numerator of 2^n*Gamma(n + 3/4)/(Gamma(3/4)*n!). - _R. J. Mathar_, Nov 06 2011
%F Numerator of integral_{x>0} 1/(x^4 + 1)^(n+1) / (Pi*sqrt(2)). - _Jean-François Alcover_, Apr 29 2013
%F From _Petros Hadjicostas_, May 23 2020: (Start)
%F 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).
%F 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)
%e 1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768, ... = A067002/A046161.
%p 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;
%t 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 *)
%Y Denominators are in A046161.
%Y Cf. A067001, A223549, A223550.
%K nonn,frac
%O 0,2
%A _N. J. A. Sloane_, Feb 16 2002
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