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%I M2508 N0992 #219 Oct 14 2023 17:49:50
%S 1,1,3,5,35,63,231,429,6435,12155,46189,88179,676039,1300075,5014575,
%T 9694845,300540195,583401555,2268783825,4418157975,34461632205,
%U 67282234305,263012370465,514589420475,8061900920775,15801325804719
%N Numerators in expansion of 1/sqrt(1-x).
%C Also numerator of binomial(2n,n)/4^n (cf. A046161).
%C Also numerator of e(n-1,n-1) (see Maple line).
%C Leading coefficient of normalized Legendre polynomial.
%C Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
%C Also the numerator of binomial(2n,n)/2^n. - _T. D. Noe_, Nov 29 2005
%C This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - _Stephen Crowley_, Apr 03 2007
%C Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - _Peter Luschny_, Aug 01 2009
%C It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - _James R. Buddenhagen_, Jan 20 2010
%C The convolution of sequence binomial(2n,n)/4^n with itself is the constant sequence with all terms = 1.
%C a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - _John M. Campbell_, Jul 04 2011
%C a(n) = denominator of 2^n*n!*n!/(2*n)!. - _Artur Jasinski_, Nov 26 2011
%C a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2x+2)^n dx. - _Leonid Bedratyuk_, Nov 17 2012
%C a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - _Jean-François Alcover_, Mar 21 2013
%C Also numerators in expansion of arcsin(x). - _Jean-François Alcover_, May 17 2013
%C Constant terms for normalized Legendre polynomials. - _Tom Copeland_, Feb 04 2016
%C From _Ralf Steiner_, Apr 07 2017: (Start)
%C By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
%C a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
%C Sum_{k>=0} a(k)/A060818(k) = -i.
%C Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3).
%C Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3).
%C a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
%C Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2).
%C Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End)
%C a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
%C Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/A344402. - _Peter Luschny_, May 20 2021
%C Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - _Nicholas Juricic_, Mar 31 2022
%C a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - _Mohammed Yaseen_, Jul 29 2023
%C a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - _Mohammed Yaseen_, Sep 19 2023
%D P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
%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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.
%H Robert G. Wilson v, <a href="/A001790/b001790.txt">Table of n, a(n) for n = 0..1666</a> (first 201 terms from T. D. Noe)
%H Horst Alzer and Bent Fuglede, <a href="http://dx.doi.org/10.1016/j.jnt.2004.12.001">Normalized binomial mid-coefficients and power means</a>, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294.
%H C. M. Bender and K. A. Milton, <a href="http://arxiv.org/abs/hep-th/9304052">Continued fraction as a discrete nonlinear transform</a>, arXiv:hep-th/9304062, 1993 (see V_n with N=1).
%H W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a> [Annotated scanned copy]
%H W. G. Bickley and J. C. P. Miller, <a href="http://dx.doi.org/10.1080/14786444208521334">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables).
%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 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%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 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H H. E. Salzer, <a href="http://dx.doi.org/10.1090/S0025-5718-1948-0023123-5">Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials</a>, Math. Comp., 3 (1948), 16-18.
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%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/LegendrePolynomial.html">Legendre Polynomial</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lorentz_factor">Lorentz Factor</a>.
%F a(n) = A000984(n)/A001316(n) where A001316(n) is the highest power of 2 dividing C(2n, n)=A000984(n). - _Benoit Cloitre_, Jan 27 2002
%F a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
%F L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0.
%F Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - _Tom Copeland_, Dec 04 2013
%F From _Ralf Steiner_, Apr 08 2017: (Start)
%F a(n)/A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)).
%F Sum_{k>=0} a(k)/A061549(k) = 2/sqrt(3).
%F Sum_{k>=0} (-1)^k*a(k)/A061549(k) = 2/sqrt(5).
%F Sum_{k>=0} (-1)^(k+1)*a(k)/A061549(k) = -2/sqrt(5).
%F a(n)/A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)).
%F Sum_{k>=0} a(k)/A123854(k) = sqrt(2).
%F Sum_{k>=0} (-1)^k*a(k)/A123854(k) = sqrt(2/3).
%F Sum_{k>=0} (-1)^(k+1)*a(k)/A123854(k) = -sqrt(2/3). (End)
%F a(n) = 2^A007814(n)*(2*n-1)*a(n-1)/n. - _John Lawrence_, Jul 17 2020
%e 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
%e binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...
%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;
%p # From _Peter Luschny_, Aug 01 2009: (Start)
%p swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
%p sigma := n -> 2^(add(i,i=convert(iquo(n,2),base,2))):
%p a := n -> swing(2*n)/sigma(2*n); # (End)
%p A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # _R. J. Mathar_, Jan 18 2013
%t Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
%t Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}] (* _John M. Campbell_, Jul 04 2011 *)
%t Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]),{n,0,20}]] (* _Ralf Steiner_, Apr 07 2017 *)
%t Numerator[Table[Binomial[2n,n]/2^n, {n, 0, 25}]] (* _Vaclav Kotesovec_, Apr 07 2017 *)
%t Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* _Andres Cicuttin_, Jan 22 2018 *)
%t A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* _John Lawrence_, Jul 17 2020 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};
%o (PARI) a(n)=binomial(2*n,n)>>hammingweight(n); \\ _Gleb Koloskov_, Sep 26 2021
%o (Sage) # uses[A000120]
%o @CachedFunction
%o def swing(n):
%o if n == 0: return 1
%o return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
%o A001790 = lambda n: swing(2*n)/2^A000120(2*n)
%o [A001790(n) for n in (0..25)] # _Peter Luschny_, Nov 19 2012
%Y Cf. A001800, A001801, A008316, A046161.
%Y First column and diagonal 1 of triangle A100258.
%Y Bisection of A036069.
%Y Cf. A005187, A060818(n)= denominator(L(n)). Bisections give A061548 and A063079.
%Y From _Johannes W. Meijer_, Jun 08 2009: (Start)
%Y Cf. A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
%Y A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n.
%Y (End)
%Y A163590 is the odd part of the swinging factorial, A001803 at odd indices.
%Y Inverse Moebius transform of A180403/A046161. - _Mats Granvik_, Sep 04, 2010
%Y Cf. A123854 (denominators), A061549 (denominators). - _Ralf Steiner_, Apr 08 2017
%K nonn,easy,nice,frac
%O 0,3
%A _N. J. A. Sloane_
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