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%I #148 Oct 15 2023 20:59:50
%S 1,2,8,16,128,256,1024,2048,32768,65536,262144,524288,4194304,8388608,
%T 33554432,67108864,2147483648,4294967296,17179869184,34359738368,
%U 274877906944,549755813888,2199023255552,4398046511104,70368744177664,140737488355328,562949953421312
%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
%C Denominators of Pochhammer(n+1, -1/2)/sqrt(Pi). - _Adam Hugill_, Sep 11 2022
%C a(n) is the denominator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - _Stefano Spezia_, May 16 2023
%C 4^n/binomial(2n,n) is the expected value of the number of socks that are randomly drawn out of a drawer of n different pairs of socks, when one sock is drawn out at a time until a matching pair is found (King and King, 2005). - _Amiram Eldar_, Jul 02 2023
%C a(n) is the denominator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding numerator is A001790(n). - _Mohammed Yaseen_, Jul 29 2023
%C a(n) is the numerator of Integral_{x=0..Pi/2} sin(x)^(2*n+1) dx. The corresponding denominator is A001803(n). - _Mohammed Yaseen_, Sep 22 2023
%D W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
%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, and 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 Jeremy and Patricia King, <a href="https://www.jstor.org/stable/3621252">Problem 89.G</a>, Problem Corner, The Mathematical Gazette, Vol. 90, No. 515 (2005), p. 314; <a href="https://www.jstor.org/stable/3621446">Solution</a>, ibid., Vol. 90, No. 517 (2006), pp. 163-164.
%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/BinomialSeries.html">Binomial Series</a>.
%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/LegendrePolynomial.html">Legendre Polynomial</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomMatrix.html">Random Matrix</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RandomWalk1-Dimensional.html">Random Walk 1-Dimensional</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%F a(n) = 2^(2*n - 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
%F a(n)/A001790(n) ~ sqrt(Pi*n) (King and King, 2005). - _Amiram Eldar_, Jul 02 2023
%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
%o (Python 3.10+)
%o def A046161(n): return 1<<(n<<1)-n.bit_count() # _Chai Wah Wu_, Nov 15 2022
%Y Cf. A001790, A001803, A002596, A005187, A067002, A072287, A142961.
%Y Cf. A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.
%Y Cf. A000108, A000120, A000466, A001511, A005563, A048881, A060818, A061550, A120778, A173294, A173296, A173755, A280442.
%K nonn,easy,nice,frac
%O 0,2
%A _Eric W. Weisstein_
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