A049331 - OEIS

%I #46 Apr 07 2023 10:51:24

%S 2,2,8,3,384,40,23040,630,1146880,72576,1857945600,3326400,

%T 108999475200,148262400,2645053931520,13621608000,457065319366656000,

%U 75277762560,33566877054287216640,243290200817664

%N Denominator of (1/Pi)*Integral_{0..oo} (sin x / x)^n dx.

%H T. D. Noe, <a href="/A049331/b049331.txt">Table of n, a(n) for n = 1..100</a>

%H Ulrich Abel and Vitaliy Kushnirevych, <a href="https://doi.org/10.1007/s00591-023-00342-5">Sinc integrals revisited</a>, Mathematische Semesterberichte (2023).

%H Iskander Aliev, <a href="http://arxiv.org/abs/math/0503115">Siegel's Lemma and Sum-Distinct Sets</a>, arXiv:math/0503115 [math.NT] (2005) and  Discrete and Computational Geometry, Volume 39, Numbers 1-3 / March, 2008. [Added by _N. J. A. Sloane_, Jul 09 2009]

%H Iskander Aliev and Martin Henk, <a href="https://arxiv.org/abs/2304.00120">Minkowski's successive minima in convex and discrete geometry</a>, arXiv:2304.00120 [math.MG], 2023.

%H R. Baillie, D. Borwein and J. M. Borwein, <a href="http://www.jstor.org/stable/27642636">Surprising Sinc Sums and Integrals</a>, Amer. Math. Monthly, 115 (2008), 888-901.

%H A. H. R. Grimsey, <a href="http://dx.doi.org/10.1080/14786444508521508">On the accumulation of chance effects and the Gaussian frequency distribution</a>, Phil. Mag., 36 (1945), 294-295.

%H R. G. Medhurst and J. H. Roberts, <a href="http://dx.doi.org/10.1090/S0025-5718-1965-0172446-8">Evaluation of the integral I_n(b) = (2/Pi)*Integral_{0..inf} (sin x / x)^n cos (bx) dx</a>, Math. Comp., 19 (1965), 113-117.

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

%F a(n) = denominator( n*A099765(n)/(2^n*(n-1)!) ). - _G. C. Greubel_, Apr 01 2022

%e 1/2, 1/2, 3/8, 1/3, 115/384, 11/40, ...

%t Table[ 1/Pi*Integrate[Sinc[x]^n, {x, 0, Infinity}] // Denominator, {n, 1, 20}] (* _Jean-François Alcover_, Dec 02 2013 *)

%t Denominator@Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}]/((n-1)! 2^n), {n, 1, 30}] (* _Vladimir Reshetnikov_, Sep 02 2016 *)

%o (Magma) [Denominator( (1/(2^n*Factorial(n-1)))*(&+[(-1)^j*Binomial(n,j)*(n-2*j)^(n-1): j in [0..Floor(n/2)]]) ): n in [1..25]]; // _G. C. Greubel_, Apr 01 2022

%o (Sage) [denominator( (1/(2^n*factorial(n-1)))*sum((-1)^j*binomial(n,j)*(n-2*j)^(n-1) for j in (0..(n//2))) ) for n in (1..25)] # _G. C. Greubel_, Apr 01 2022

%Y Cf. 2*A002298 (except for n=4 term), A049330.

%Y Cf. A002304, A002305.

%K nonn,frac,easy,nice

%O 1,1

%A _N. J. A. Sloane_, Mark S. Riggs (msr1(AT)ra.msstate.edu), Dec 11 1999