A049330
Numerator of (1/Pi)*Integral_{x=0..oo} (sin(x)/x)^n dx.
Original entry on oeis.org
1, 1, 3, 1, 115, 11, 5887, 151, 259723, 15619, 381773117, 655177, 20646903199, 27085381, 467168310097, 2330931341, 75920439315929441, 12157712239, 5278968781483042969, 37307713155613, 9093099984535515162569, 339781108897078469, 168702835448329388944396777
Offset: 1
1/2, 1/2, 3/8, 1/3, 115/384, 11/40, ...
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.3, p. 22.
- 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, (2005) arXiv:math/0503115 [math.NT]; 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.
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[Numerator( (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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Numerator[Table[Integrate[(Sin[x]/x)^n,{x,0,\[Infinity]}]/Pi,{n,25}]] (* Harvey P. Dale, Jan 01 2013 *)
Numerator@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 *)
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[numerator( (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
A002298
Denominator of (2/Pi)*Integral_{0..inf} (sin x / x)^n dx.
Original entry on oeis.org
1, 1, 4, 3, 192, 20, 11520, 315, 573440, 36288, 928972800, 1663200, 54499737600, 74131200, 1322526965760, 6810804000, 228532659683328000, 37638881280, 16783438527143608320, 121645100408832, 30370031620545576960000
Offset: 1
1, 1, 3/4, 2/3, 115/192, 11/20, ...
- 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).
- T. D. Noe, Table of n, a(n) for n=1..100
- 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.
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Denominator[Table[2/Pi Integrate[(Sin[x]/x)^n,{x,0,\[Infinity]}],{n,25}]] (* Harvey P. Dale, Sep 04 2011 *)
Denominator@Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}]/((n-1)! 2^(n-1)), {n, 1, 30}] (* Vladimir Reshetnikov, Sep 02 2016 *)
Corrected and extended by Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 02 2001
A002304
Numerators of coefficients in asymptotic expansion of (2/Pi)*Integral_{0..oo} (sin x / x)^n dx.
Original entry on oeis.org
1, -3, -13, 27, 52791, 482427, -124996631, -5270328789, -7479063506161, 6921977624613, 10703530420192887741, -31023547697719285017327, 4502691897987538544182239, -201974203900639732887399429, 632827656013898657214770949567, -1732419272534268233524732551
Offset: 0
- 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).
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nmax = 20; Numerator[CoefficientList[Simplify[Sum[3^k*(2*k)!/(k!*2^k*n^k) * SeriesCoefficient[Exp[n*(x^2/6 + Sum[(-1)^m*BernoulliB[2*m]* 2^(2*m - 1)*(x^(2*m)/(m*(2*m)!)), {m, 1, k}])], {x, 0, 2*k}], {k, 0, nmax}]], 1/n]] (* Vaclav Kotesovec, Aug 10 2019 *)
A002305
Denominators of coefficients in asymptotic expansion of (2/Pi)*Integral_{0..oo} (sin x / x)^n dx.
Original entry on oeis.org
1, 20, 1120, 3200, 3942400, 66560000, 10035200000, 136478720000, 268461670400000, 56518246400000, 23658537943040000000, 51431604224000000, 70718455808000000, 102541760921600000, 23292891381760000000, 8879987916800000, 144993552704000000, 1072952290009600000
Offset: 0
- 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).
- David H. Bailey and Jonathan M. Borwein, Experimental computation with oscillatory integrals, Gems in experimental mathematics, 25-40, Contemp. Math., 517, Amer. Math. Soc., Providence, RI, 2010. [Added by _N. J. A. Sloane_, Nov 02 2009]
- 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.
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nmax = 20; Denominator[CoefficientList[Simplify[Sum[3^k*(2*k)!/(k!*2^k*n^k) * SeriesCoefficient[Exp[n*(x^2/6 + Sum[(-1)^m*BernoulliB[2*m]* 2^(2*m - 1)*(x^(2*m)/(m*(2*m)!)), {m, 1, k}])], {x, 0, 2*k}], {k, 0, nmax}]], 1/n]] (* Vaclav Kotesovec, Aug 10 2019 *)
Showing 1-4 of 4 results.
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