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
A176366
Denominator of (1/Pi)*Integral_{0..infinity} (sin x / x)^(2*n) dx.
Original entry on oeis.org
2, 3, 40, 630, 72576, 3326400, 148262400, 13621608000, 75277762560, 243290200817664, 2322315553259520000, 538583682060103680000, 85226428809510912000000, 27777728189842735104000000, 147362699895661699242393600000, 4282728465717668134232064000000
Offset: 1
a(2) = 3 because Integral_{0..infinity} (sin(x)/x)^4 dx = (1/3)*Pi.
a(3) = 40 because Integral_{0..infinity} (sin(x)/x)^6 dx = (11/40)*Pi.
a(4) = 630 because Integral_{0..infinity} (sin(x)/x)^8 dx = (151/630)*Pi.
a(5) = 72576 because Integral_{0..infinity} (sin(x)/x)^10 dx = (15619/72576)*Pi.
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a[n_]:= (1/Pi)*Integrate[(Sin[x]/x)^(2n), {x, 0, Infinity}]//Denominator;
Array[a, 16] (* Jean-François Alcover, Nov 25 2017 *)
A261398
Integer coefficients arising from a formula for Sum_{m>=1} sin(Pi*m/3)^2/m^2.
Original entry on oeis.org
1, 2, 6, 32, 230, 2112, 23548, 309248, 4675014, 79969280, 1527092468, 32203259904, 743288515164, 18638209056768, 504541774904760, 14664951970922496, 455522635895576646, 15058911973677465600, 527896878148304296900, 19559986314930028544000, 763820398700983273655796, 31353195811771939838492672
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- R. Butler, On the Evaluation of Integral_{x=0..oo} (sin(t))^m/t^m dt by the Trapezoidal Rule, The American Mathematical Monthly, vol. 67, no. 6, 1960, pp. 566-69.
- J. W. H. Swanepoel, On a generalization of a theorem by Euler, Journal of Number Theory 149 (2015) 46-56.
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[(&+[(-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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A261398 := proc(n)
add( (-1)^i*binomial(n,i)*(n-2*i)^(n-1),i=0..floor((n-1)/2)) ;
end proc:
seq(A261398(n),n=1..25) ; # R. J. Mathar, Aug 19 2015
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Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}], {n, 1, 20}] (* Vladimir Reshetnikov, Sep 05 2016 *)
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a(n) = sum(i=0, (n-1)\2, (-1)^i*binomial(n,i)*(n-2*i)^(n-1)); \\ Michel Marcus, Sep 05 2016
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[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
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