A049331 Denominator of (1/Pi)*Integral_{0..oo} (sin x / x)^n dx.
2, 2, 8, 3, 384, 40, 23040, 630, 1146880, 72576, 1857945600, 3326400, 108999475200, 148262400, 2645053931520, 13621608000, 457065319366656000, 75277762560, 33566877054287216640, 243290200817664
Offset: 1
Examples
1/2, 1/2, 3/8, 1/3, 115/384, 11/40, ...
References
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.
Links
- T. D. Noe, Table of n, a(n) for n = 1..100
- Ulrich Abel and Vitaliy Kushnirevych, Sinc integrals revisited, Mathematische Semesterberichte (2023).
- Iskander Aliev, Siegel's Lemma and Sum-Distinct Sets, 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]
- Iskander Aliev and Martin Henk, Minkowski's successive minima in convex and discrete geometry, arXiv:2304.00120 [math.MG], 2023.
- Robert Baillie, David Borwein, and Jonathan M. Borwein, Surprising Sinc Sums and Integrals, Amer. Math. Monthly, 115 (2008), 888-901.
- A. H. R. Grimsey, On the accumulation of chance effects and the Gaussian frequency distribution, Phil. Mag., 36 (1945), 294-295.
- R. G. Medhurst and J. H. Roberts, Evaluation of the integral I_n(b) = (2/Pi)*Integral_{0..inf} (sin x / x)^n cos (bx) dx, Math. Comp., 19 (1965), 113-117.
- Jan W. H. Swanepoel, A Short Simple Probabilistic Proof of a Well Known Identity and the Derivation of Related New Identities Involving the Bernoulli Numbers and the Euler Numbers, Integers (2025) Vol. 25, Art. No. A50. See p. 1.
- Eric Weisstein's World of Mathematics, Sinc Function.
Programs
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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
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Mathematica
Table[ 1/Pi*Integrate[Sinc[x]^n, {x, 0, Infinity}] // Denominator, {n, 1, 20}] (* Jean-François Alcover, Dec 02 2013 *) 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 *) -
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
Formula
a(n) = denominator( n*A099765(n)/(2^n*(n-1)!) ). - G. C. Greubel, Apr 01 2022