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A049331
Denominator of (1/Pi)*Integral_{0..oo} (sin x / x)^n dx.
5
2, 2, 8, 3, 384, 40, 23040, 630, 1146880, 72576, 1857945600, 3326400, 108999475200, 148262400, 2645053931520, 13621608000, 457065319366656000, 75277762560, 33566877054287216640, 243290200817664
OFFSET
1,1
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
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.
R. Baillie, D. Borwein, and J. M. Borwein, Surprising Sinc Sums and Integrals, Amer. Math. Monthly, 115 (2008), 888-901.
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.
Eric Weisstein's World of Mathematics, Sinc Function.
FORMULA
a(n) = denominator( n*A099765(n)/(2^n*(n-1)!) ). - G. C. Greubel, Apr 01 2022
EXAMPLE
1/2, 1/2, 3/8, 1/3, 115/384, 11/40, ...
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 *)
PROG
(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
(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
CROSSREFS
Cf. 2*A002298 (except for n=4 term), A049330.
Sequence in context: A093731 A195361 A274041 * A369771 A239677 A331333
KEYWORD
nonn,frac,easy,nice,changed
AUTHOR
N. J. A. Sloane, Mark S. Riggs (msr1(AT)ra.msstate.edu), Dec 11 1999
STATUS
approved