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A001803 Numerators in expansion of (1 - x)^(-3/2).

%I A001803 M2986 N1207 #116 Apr 30 2025 16:56:16
%S A001803 1,3,15,35,315,693,3003,6435,109395,230945,969969,2028117,16900975,
%T A001803 35102025,145422675,300540195,9917826435,20419054425,83945001525,
%U A001803 172308161025,1412926920405,2893136075115,11835556670925
%N A001803 Numerators in expansion of (1 - x)^(-3/2).
%C A001803 a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). - _James R. Buddenhagen_, Aug 17 2008
%C A001803 a(n) is the denominator of (2n)!!/(2*n + 1)!! = 2^(2*n)*n!*n!/(2*n + 1)! (see Andersson). - _N. J. A. Sloane_, Jun 27 2011
%C A001803 a(n) = (2*n + 1)*A001790(n). A046161(n)/a(n) = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (i.e., 1 - 1/3 + 1/5 - 1/7 + ... ). See A173384 and A173396. - _Paul Curtz_, Feb 21 2010
%C A001803 a(n) is the denominator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding numerator is A101926(n). - _Mohammed Yaseen_, Jul 25 2023
%D A001803 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
%D A001803 G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
%D A001803 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001803 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001803 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.
%H A001803 T. D. Noe, <a href="/A001803/b001803.txt">Table of n, a(n) for n = 0..200</a>
%H A001803 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A001803 Mats Erik Andersson, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa85/aa8542.pdf">Das Flaviussche Sieb</a>, Acta Arith., 85 (1998), 301-307.
%H A001803 Alexander Barg, <a href="https://arxiv.org/abs/2005.12995">Stolarsky's invariance principle for finite metric spaces</a>, arXiv:2005.12995 [math.CO], 2020.
%H A001803 Yue-Wu Li and Feng Qi, <a href="https://doi.org/10.3390/axioms13050317">A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments</a>, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
%H A001803 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A001803 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Heads-Minus-TailsDistribution.html">Heads-Minus-Tails Distribution</a>, <a href="https://mathworld.wolfram.com/RandomWalk1-Dimensional.html">Random Walk--1-Dimensional</a>, <a href="https://mathworld.wolfram.com/CircleLinePicking.html">Circle Line Picking</a>.
%F A001803 a(n) = (2*n + 1)! /(n!^2*2^A000120(n)) = (n + 1)*binomial(2*n+2,n+1)/2^(A000120(n)+1). - _Ralf Stephan_, Mar 10 2004
%F A001803 From _Johannes W. Meijer_, Jun 08 2009: (Start)
%F A001803 a(n) = numerator( (2*n+1)*binomial(2*n,n)/(4^n) ).
%F A001803 (1 - x)^(-3/2) = Sum_{n>=0} ((2*n+1)*binomial(2*n,n)/4^n)*x^n. (End)
%F A001803 Truncations of rational expressions like those given by the numerator or denominator operators 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+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - _Peter Luschny_, Aug 01 2009
%F A001803 a(n) = denominator(Pi*binomial(n, -1/2)). - _Peter Luschny_, Dec 06 2024
%p A001803 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 A001803 sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
%p A001803 a := n -> swing(2*n+1)/sigma(2*n+1); # _Peter Luschny_, Aug 01 2009
%p A001803 A001803 := proc(n) (2*n+1)*binomial(2*n,n)/4^n ; numer(%) ; end proc: # _R. J. Mathar_, Jul 06 2011
%p A001803 a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # _Peter Luschny_, Dec 06 2024
%t A001803 Numerator/@CoefficientList[Series[(1-x)^(-3/2),{x,0,25}],x]  (* _Harvey P. Dale_, Feb 19 2011 *)
%t A001803 Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* _Gerry Martens_, Nov 13 2016 *)
%o A001803 (PARI) a(n) = numerator((2*n+1)*binomial(2*n,n)/(4^n)); \\ _Altug Alkan_, Sep 06 2018
%o A001803 (Julia)
%o A001803 A001803(n) = sum(<<(A001790(k), A005187(n) - A005187(k)) for k in 0:n) # _Peter Luschny_, Oct 03 2019
%o A001803 (Magma)
%o A001803 A001803:= func< n | Numerator(Binomial(n+2,2)*Catalan(n+1)/4^n) >;
%o A001803 [A001803(n): n in [0..30]]; // _G. C. Greubel_, Apr 27 2025
%o A001803 (SageMath)
%o A001803 def A001803(n): return numerator((n+1)*binomial(2*n+2,n+1)/2^(2*n+1))
%o A001803 print([A001803(n) for n in range(31)]) # _G. C. Greubel_, Apr 27 2025
%Y A001803 The denominator is given in A046161.
%Y A001803 Largest odd divisors of A001800, A002011, A002457, A005430, A033876, A086228.
%Y A001803 Bisection of A004731, A004735, A086116.
%Y A001803 Second column of triangle A100258.
%Y A001803 Cf. A161199, A161201, A173384, A173396.
%Y A001803 Cf. A002596 (numerators in expansion of (1-x)^(1/2)).
%Y A001803 Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)).
%Y A001803 A163590 is the odd part of the swinging factorial, A001790 at even indices. - _Peter Luschny_, Aug 01 2009
%K A001803 nonn,frac
%O A001803 0,2
%A A001803 _N. J. A. Sloane_