Search: id:A046161
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%I A046161
%S A046161 1,2,8,16,128,256,1024,2048,32768,65536,262144,524288,4194304,8388608,
%T A046161 33554432,67108864,2147483648,4294967296,17179869184,34359738368,
%U A046161 274877906944,549755813888,2199023255552,4398046511104,70368744177664
%N A046161 Denominator of binomial(2n,n)/4^n.
%C A046161 Also denominator of e(0,n) (see Maple line). - N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 16 2002.
%C A046161 Denominator of coefficient of x^n in (1+x)^(k/2) or (1-x)^(k/2) for any 
               odd integer k. - Michael Somos, Sep 15 2004
%C A046161 a(n)=4^n/2^A000120(n). - Michael Somos, Sep 15 2004
%C A046161 Numerator of binomial(2n,n)/4^n = A001790(n).
%C A046161 Denominators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - 
               Paul Barry (pbarry(AT)wit.ie), Jul 12 2005
%C A046161 Denominator of 2^m*GAMMA(m+3/4)/(GAMMA(3/4)*GAMMA(m+1)). - Stephen Crowley 
               (crow(AT)crowlogic.net), Mar 19 2007
%C A046161 Denominator in expansion of Jacobi_P(n,1/2,1/2,x). - Paul Barry (pbarry(AT)wit.ie), 
               Feb 13 2008
%C A046161 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%C A046161 This sequence equals the denominators of the coefficients of the series 
               expansions of (1-x)^((-1-2*n)/2) for all integer values of n; see 
               A161198 for detailed information.
%C A046161 . (End)
%D A046161 B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 
               1, p. 513, Eq. (7.282).
%D A046161 V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. 
               Math. Soc., 49 (No. 3, March 2002), 311-317.
%H A046161 T. D. Noe, <a href="b046161.txt">Table of n, a(n) for n=0..200</a>
%H A046161 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Heads-Minus-TailsDistribution.html">Link to a section of The World 
               of Mathematics.</a>
%H A046161 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RandomWalk1-Dimensional.html">Link to a section of The World of Mathematics.</
               a>
%H A046161 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LegendrePolynomial.html">Legendre Polynomial</a>
%H A046161 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinomialSeries.html">Binomial Series</a>
%H A046161 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RandomMatrix.html">Random Matrix</a>
%F A046161 a(n)=2^(2n-1-A048881(n-1)), if n>0.
%e A046161 sqrt(1+x) = 1+1/2*x-1/8*x^2+1/16*x^3-5/128*x^4+7/256*x^5-21/1024*x^6+33/
               2048*x^7+...
%e A046161 binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/
               2048, 6435/32768, ...
%e A046161 The sequence e(0,n) begins 1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/
               1024, 129789/2048, 4023459/32768 ...
%p A046161 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;
%p A046161 Z[0]:=0: for k to 30 do Z[k]:=simplify(1/(2-z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), 
               j=1..30): gser:=series(g, z=0, 27): seq(denom(coeff(gser, z, n)), 
               n=-1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 21 
               2008
%t A046161 a[n_, m_] := Binomial[n - m/2 + 1, n - m + 1] - Binomial[n - m/2, n - 
               m + 1]; s[n_] := Sum[ a[n, k], {k, 0, n}]; Table [Denominator[s[n]], 
               {n, 0, 26}] - Michele Dondi (bik.mido(AT)tiscalinet.it), Jul 11, 
               2002
%o A046161 (PARI) a(n)=if(n<0,0,denominator(binomial(2*n,n)/4^n)) /* Michael Somos, 
               Sep 15 2004 */
%Y A046161 Cf. A001790, A001803, A002596, A005187, A072287, A067002.
%Y A046161 a(n) = 2^A005187(n).
%Y A046161 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 
               2009: (Start)
%Y A046161 Cf. A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/
               2) for all values of n.
%Y A046161 (End)
%Y A046161 Sequence in context: A098232 A100736 A099888 this_sequence A092978 A101059 
               A101658
%Y A046161 Adjacent sequences: A046158 A046159 A046160 this_sequence A046162 A046163 
               A046164
%K A046161 nonn,easy,nice,frac
%O A046161 0,2
%A A046161 Eric Weisstein (eric(AT)weisstein.com)

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