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A067002 Numerator of sum_{k=0..n} 2^(k-2n) *binomial(2n-2k,n-k)* binomial(n+k,n). 2

%I

%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-2n) *binomial(2n-2k,n-k)* binomial(n+k,n).

%C Numerator of e(0,n) (see 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-Fran├žois Alcover_, Apr 29 2013]

%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.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_, Feb 16 2002

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Last modified February 21 01:15 EST 2019. Contains 320364 sequences. (Running on oeis4.)