A049330 -id:A049330 - cp's OEIS frontend

A002297 Numerator of (2/Pi)*Integral_{0..inf} (sin x / x)^n dx.

1, 1, 3, 2, 115, 11, 5887, 151, 259723, 15619, 381773117, 655177, 20646903199, 27085381, 467168310097, 2330931341, 75920439315929441, 12157712239, 5278968781483042969, 37307713155613, 9093099984535515162569, 339781108897078469, 168702835448329388944396777

Offset: 1

N. J. A. Sloane

			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.

Cf. A002298 (for denominators), A002304, A002305. Essentially the same as A049330, except for the n=4 term.

a[n_] := Numerator[ (2/Pi)*Integrate[ (Sin[x]/x)^n, {x, 0, Infinity}] ]; Table[ a[n], {n, 1, 21}] (* Jean-François Alcover, Dec 19 2011 *)
Numerator@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 *)

a(n) = numerator((n/2^(n-1)) * sum(r=0, n/2, (-1)^r*(n-2*r)^(n-1)/(r!*(n-r)!))); \\ Michel Marcus, Oct 02 2013

a(n) = numerator((n/2^(n-1)) * sum((-1)^r*(n-2*r)^(n-1)/(r!*(n-r)!), r=0..n/2)). - Sean A. Irvine, Oct 01 2013

a(22)-a(23) from T. D. Noe, Feb 22 2014

A049331 Denominator of (1/Pi)*Integral_{0..oo} (sin x / x)^n dx.

Original entry on oeis.org

2, 2, 8, 3, 384, 40, 23040, 630, 1146880, 72576, 1857945600, 3326400, 108999475200, 148262400, 2645053931520, 13621608000, 457065319366656000, 75277762560, 33566877054287216640, 243290200817664

Offset: 1

N. J. A. Sloane, Mark S. Riggs (msr1(AT)ra.msstate.edu), Dec 11 1999

			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, 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.

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

Cf. A002304, A002305.

[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

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 *)

[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

a(n) = denominator( n*A099765(n)/(2^n*(n-1)!) ). - G. C. Greubel, Apr 01 2022

A099765 a(n) = (1/Pi)(2^n/n)(n-1)!*Integral_{t>=0} (sin(t)/t)^n dt.

Original entry on oeis.org

1, 1, 2, 8, 46, 352, 3364, 38656, 519446, 7996928, 138826588, 2683604992, 57176039628, 1331300646912, 33636118326984, 916559498182656, 26795449170328038, 836606220759859200, 27784046218331805100

Offset: 1

Benoit Cloitre, Nov 11 2004, Dec 11 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..406
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010. [From _N. J. A. Sloane_, Sep 27 2010]
W. Trump, Magic series.
Eric Weisstein's World of Mathematics, Sinc Function

Cf. A049330, A261398.

[(1/n)*(&+[(-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

Table[1/n Sum[(-1)^k Binomial[n,k](n-2k)^(n-1),{k,0,Floor[n/2]}], {n,20}] (* Harvey P. Dale, Oct 21 2011 *)

a(n)=(1/n)*sum(k=0,floor(n/2),(-1)^k*binomial(n,k)*(n-2*k)^(n-1))

[(1/n)*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

a(n) = (1/n) * Sum_{k=0, floor(n/2)} (-1)^k * binomial(n, k) * (n-2*k)^(n-1).

a(n) = A261398(n)/n. - Vladimir Reshetnikov, Sep 05 2016

A176365 Numerator of (1/Pi)Integral_{0..infinity} (sin x / x)^(2n) dx.

Original entry on oeis.org

1, 1, 11, 151, 15619, 655177, 27085381, 2330931341, 12157712239, 37307713155613, 339781108897078469, 75489558096433522049, 11482547005345338463969, 3607856726470666022715979, 18497593486903125823791655511, 520679973964725199436393399689

Offset: 1

Jonathan Vos Post, Apr 16 2010

The denominators are given in A176366.

Bisection of A049330. See it for further references.

			a(2) = 1 because Integral_{0..infinity} (sin(x)/x)^4 dx = (1/3)*Pi.
a(3) = 11 because Integral_{0..infinity} (sin(x)/x)^6 dx = (11/40)*Pi.
a(4) = 151 because Integral_{0..infinity} (sin(x)/x)^8 dx = (151/630)*Pi.
a(5) = 15619 because Integral_{0..infinity} (sin(x)/x)^10 dx = (15619/72576)*Pi.

G. C. Greubel, Table of n, a(n) for n = 1..75
M. R. Darafsheh, Hassan Jolany, An extension of Lobachevsky formula, arXiv:1004.2653 [math.GM], 2010-2017.

Cf. A049330, A176366.

A176365 := proc(n) sin(x)^(2*n)/x^(2*n) ; int(%,x=0..infinity)/Pi ; numer(%) ; end proc: # R. J. Mathar, Apr 24 2010

a[n_]:= (1/Pi)*Integrate[(Sin[x]/x)^(2n), {x, 0, Infinity}]//Numerator; Array[a, 16] (* Jean-François Alcover, Nov 25 2017 *)

a(n) = A049330(2*n).

5 terms added and broken URL corrected by R. J. Mathar, Apr 24 2010

Further terms from Max Alekseyev, May 07 2010

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

N. J. A. Sloane, Aug 18 2015

nonn

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.

Cf. A049330, A049331, A099765.

[(&+[(-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

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

Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}], {n, 1, 20}] (* Vladimir Reshetnikov, Sep 05 2016 *)

a(n) = sum(i=0, (n-1)\2, (-1)^i*binomial(n,i)*(n-2*i)^(n-1)); \\ Michel Marcus, Sep 05 2016

[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

a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i*binomial(n,i)*(n-2*i)^(n-1).
a(n)/(2^n*(n-1)!) = A049330(n)/A049331(n).
a(n) = n * A099765(n). - Vladimir Reshetnikov, Sep 05 2016

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A002297 Numerator of (2/Pi)*Integral_{0..inf} (sin x / x)^n dx.

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A049331 Denominator of (1/Pi)*Integral_{0..oo} (sin x / x)^n dx.

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A099765 a(n) = (1/Pi)*(2^n/n)*(n-1)!*Integral_{t>=0} (sin(t)/t)^n dt.

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A176365 Numerator of (1/Pi)*Integral_{0..infinity} (sin x / x)^(2*n) dx.

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A261398 Integer coefficients arising from a formula for Sum_{m>=1} sin(Pi*m/3)^2/m^2.

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A099765 a(n) = (1/Pi)(2^n/n)(n-1)!*Integral_{t>=0} (sin(t)/t)^n dt.

A176365 Numerator of (1/Pi)Integral_{0..infinity} (sin x / x)^(2n) dx.