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 A001790 Numerators in expansion of 1/sqrt(1-x). (Formerly M2508 N0992) 68
 1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also numerator of binomial(2n,n)/4^n (cf. A046161). Also numerator of e(n-1,n-1) (see Maple line). Leading coefficient of normalized Legendre polynomial. Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x). Also the numerator of binomial(2n,n)/2^n. - T. D. Noe, Nov 29 2005 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 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 It appears that a(n) = A060818(n)*A001147(n)/A000142(n). - James R. Buddenhagen, Jan 20 2010 The convolution of sequence binomial(2n,n)/4^n with itself is the constant sequence with all terms = 1. a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011 a(n) = denominator of 2^n*n!*n!/(2*n)!. - Artur Jasinski, Nov 26 2011 a(n) = numerator of (1/Pi)*Integral_{x=-infinity..+infinity} 1/(x^2-2x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012 a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013 Also numerators in expansion of arcsin(x). - Jean-François Alcover, May 17 2013 Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016 From Ralf Steiner, Apr 07 2017: (Start) By analytic continuation to the entire complex plane there exist regularized values for divergent sums: a(n)/A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} a(k)/A060818(k) = -i. Sum_{k>=0} (-1)^k*a(k)/A060818(k) = 1/sqrt(3). Sum_{k>=0} (-1)^(k+1)*a(k)/A060818(k) = -1/sqrt(3). a(n)/A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} (-1)^k*a(k)/A046161(k) = 1/sqrt(2). Sum_{k>=0} (-1)^(k+1)*a(k)/A046161(k) = -1/sqrt(2). (End) a(n) = numerator of (1/Pi)*Integral_{x=-infinity..+infinity} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017 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 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 REFERENCES P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102. LINKS Robert G. Wilson v, Table of n, a(n) for n = 0..1666 (first 201 terms from T. D. Noe) Horst Alzer and Bent Fuglede, Normalized binomial mid-coefficients and power means, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 284-294. C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993 (see V_n with N=1). W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table [Annotated scanned copy] W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables). Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4. Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011. V. H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317. Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7. H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18. J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) Eric Weisstein's World of Mathematics, Binomial Series Eric Weisstein's World of Mathematics, Legendre Polynomial Wikipedia, Lorentz Factor. FORMULA 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 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). L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0. 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 From Ralf Steiner, Apr 08 2017: (Start) a(n)/A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} a(k)/A061549(k) = 2/sqrt(3). Sum_{k>=0} (-1)^k*a(k)/A061549(k) = 2/sqrt(5). Sum_{k>=0} (-1)^(k+1)*a(k)/A061549(k) = -2/sqrt(5). a(n)/A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)). Sum_{k>=0} a(k)/A123854(k) = sqrt(2). Sum_{k>=0} (-1)^k*a(k)/A123854(k) = sqrt(2/3). Sum_{k>=0} (-1)^(k+1)*a(k)/A123854(k) = -sqrt(2/3). (End) a(n) = 2^A007814(n)*(2*n-1)*a(n-1)/n. - John Lawrence, Jul 17 2020 EXAMPLE 1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ... binomial(2n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ... MAPLE 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; # From Peter Luschny, Aug 01 2009: (Start) 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: sigma := n -> 2^(add(i, i=convert(iquo(n, 2), base, 2))): a := n -> swing(2*n)/sigma(2*n); # (End) A001790 := proc(n) binomial(2*n, n)/4^n ; numer(%) ; end proc : # R. J. Mathar, Jan 18 2013 MATHEMATICA Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]] Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}]   (* John M. Campbell, Jul 04 2011 *) Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]), {n, 0, 20}]] (* Ralf Steiner, Apr 07 2017 *) Numerator[Table[Binomial[2n, n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *) Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *) 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 *) PROG (PARI) {a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))}; (PARI) a(n)=binomial(2*n, n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021 (Sage) # uses[A000120] @CachedFunction def swing(n):     if n == 0: return 1     return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n A001790 = lambda n: swing(2*n)/2^A000120(2*n) [A001790(n) for n in (0..25)]  # Peter Luschny, Nov 19 2012 CROSSREFS Cf. A001800, A001801, A008316, A046161. First column and diagonal 1 of triangle A100258. Bisection of A036069. Cf. A005187, A060818(n)= denominator(L(n)). Bisections give A061548 and A063079. From Johannes W. Meijer, Jun 08 2009: (Start) Cf. A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)]. A161198 triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n. (End) A163590 is the odd part of the swinging factorial, A001803 at odd indices. Inverse Moebius transform of A180403/A046161. - Mats Granvik, Sep 04, 2010 Cf. A123854 (denominators), A061549 (denominators). - Ralf Steiner, Apr 08 2017 Sequence in context: A259853 A052468 A055786 * A173092 A057908 A120828 Adjacent sequences:  A001787 A001788 A001789 * A001791 A001792 A001793 KEYWORD nonn,easy,nice,frac AUTHOR STATUS approved

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Last modified October 1 16:19 EDT 2022. Contains 357149 sequences. (Running on oeis4.)