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 A046161 a(n) = denominator of binomial(2n,n)/4^n. 85

%I

%S 1,2,8,16,128,256,1024,2048,32768,65536,262144,524288,4194304,8388608,

%T 33554432,67108864,2147483648,4294967296,17179869184,34359738368,

%U 274877906944,549755813888,2199023255552,4398046511104,70368744177664

%N a(n) = denominator of binomial(2n,n)/4^n.

%C Also denominator of e(0,n) (see Maple line). - _N. J. A. Sloane_, Feb 16 2002

%C 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 Numerator of binomial(2n,n)/4^n = A001790(n).

%C Denominators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - _Paul Barry_, Jul 12 2005

%C Denominator of 2^m*GAMMA(m+3/4)/(GAMMA(3/4)*GAMMA(m+1)). - _Stephen Crowley_, Mar 19 2007

%C Denominator in expansion of Jacobi_P(n,1/2,1/2,x). - _Paul Barry_, Feb 13 2008

%C 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. - _Johannes W. Meijer_, Jun 08 2009

%C Numerators of binomial transform of 1, -1/3, 1/5, -1/7, 1/9, ... (Madhava-Gregory-Leibniz series for Pi/4): 1, 2/3, 8/15, 16/35, 128/315, 256/693, .... First differences are -1/3, -2/15, -8/105, -16/315, -128/3465, -256/9009, ... which contain the same numerators, negated. The second differences are 1/5, 2/35, 8/315, 16/1155, 128/15015, ... again with the same numerators. Second column: 2/3, -2/15, 2/35, -2/63, 2/99; see A000466(n+1) = A005563(2n+1). Third column: 8*(1/15, -1/105, 1/315, -1/693, ), see A061550. See A173294 and A173296. - _Paul Curtz_, Feb 16 2010

%C 0, 1, 5/3, 11/5, 93/35, 193/63, 793/231, ... = (0 followed by A120778(n))/A001790(n) is the binomial transform of 0, 1, -1/3, 1/5, -1/7, 1/9, ... . See A173755 and formula below. - _Paul Curtz_, Mar 13 2013

%C Numerator of power series of arcsin(x)/sqrt(1-x^2), centered at x=0. - _John Molokach_, Aug 02 2013

%C Denominators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * euler(2*n)*x^n/(2*n)), see A280442 for the numerators. - _Johannes W. Meijer_, Jan 05 2017

%D B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.282).

%D Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 72.

%H T. D. Noe, <a href="/A046161/b046161.txt">Table of n, a(n) for n = 0..200</a>

%H C. M. Bender and K. A. Milton, <a href="http://arxiv.org/abs/hep-th/9304062">Continued fraction as a discrete nonlinear transform</a>, arXiv:hep-th/9304062, 1993. See V_n with N=1.

%H Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, Graça Tomaz, <a href="https://www.emis.de/journals/JIS/VOL21/Falcao/falcao2.html">Combinatorial Identities Associated with a Multidimensional Polynomial Sequence</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.

%H V. H. Moll, <a href="http://www.ams.org/notices/200203/fea-moll.pdf">The evaluation of integrals: a personal story</a>, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heads-Minus-TailsDistribution.html">Heads-Minus-Tails Distribution</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomWalk1-Dimensional.html">Random Walk 1-Dimensional</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSeries.html">Binomial Series</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomMatrix.html">Random Matrix</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%F a(n) = 2^(2n - 1 - A048881(n-1)), if n > 0.

%F a(n) = 2^A005187(n).

%F a(n) = 4^n/2^A000120(n). - _Michael Somos_, Sep 15 2004

%F a(n) = 2^A001511(n)*a(n-1) with a(0) = 1. - _Johannes W. Meijer_, Nov 04 2012

%F a(n) = denominator(binomial(-1/2,n)). - _Peter Luschny_, Nov 21 2012

%F a(n) = (0 followed by A120778(n)) + A001790(n). - _Paul Curtz_, Mar 13 2013

%F a(n) = 2^n*A060818(n). - _Johannes W. Meijer_, Jan 05 2017

%e 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 binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...

%e The sequence e(0,n) begins 1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768 ...

%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: seq(denom(e(0,n)), n = 0..24);

%p 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_, May 21 2008

%p A046161 := proc(n) option remember: if n = 0 then 1 else 2^A001511(n) * procname(n-1) fi: end: A001511 := proc(n): padic[ordp](2*n, 2) end: seq(A046161(n), n = 0..24); # _Johannes W. Meijer_, Nov 04 2012

%p A046161 := n -> 4^n/2^add(i,i=convert(n, base, 2)):

%p seq(A046161(n), n=0..24); # _Peter Luschny_, Apr 08 2014

%t 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 *)

%t Denominator[Table[Binomial[2n,n]/4^n,{n,0,30}]] (* _Harvey P. Dale_, Oct 29 2012 *)

%t Table[Denominator@LegendreP[2n,0],{n,0,24}] (* _Andres Cicuttin_, Jan 22 2018 *)

%o (PARI) a(n)=if(n<0,0,denominator(binomial(2*n,n)/4^n)) /* _Michael Somos_, Sep 15 2004 */

%o (PARI) a(n)=my(s=n);while(n>>=1,s+=n);2^s \\ _Charles R Greathouse IV_, Apr 07 2012

%o (PARI) a(n)=denominator(I^-n*pollegendre(n,I/2)) \\ _Charles R Greathouse IV_, Mar 18 2017

%o (Sage)

%o def A046161(n):

%o A005187 = lambda n: A005187(n//2) + n if n > 0 else 0

%o return 2^A005187(n)

%o [A046161(n) for n in (0..24)] # _Peter Luschny_, Nov 16 2012

%o (Maxima)

%o a(n) := denom(binomial(-1/2,n));

%o makelist(a(n),n,0,24); /* _Peter Luschny_, Nov 21 2012 */

%o (MAGMA) [Denominator(Binomial(2*n,n)/4^n): n in [0..30]]; // _Vincenzo Librandi_, Jul 18 2015

%Y Cf. A001790, A001803, A002596, A005187, A072287, A067002.

%Y Cf. A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.

%K nonn,easy,nice,frac

%O 0,2

%A _Eric W. Weisstein_

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Last modified January 21 11:54 EST 2019. Contains 319356 sequences. (Running on oeis4.)