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.
-
[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 *)
-
[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
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
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.
-
[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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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 *)
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[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
A155467
Triangle T(n, k) = Eulerian(n+1, k)*Binomial(n+1, k)/(k+1), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 6, 1, 1, 22, 22, 1, 1, 65, 220, 65, 1, 1, 171, 1510, 1510, 171, 1, 1, 420, 8337, 21140, 8337, 420, 1, 1, 988, 40068, 218666, 218666, 40068, 988, 1, 1, 2259, 175296, 1852914, 3935988, 1852914, 175296, 2259, 1, 1, 5065, 717600, 13655760, 55034868, 55034868, 13655760, 717600, 5065, 1
Offset: 0
Triangle begins as:
1;
1, 1;
1, 6, 1;
1, 22, 22, 1;
1, 65, 220, 65, 1;
1, 171, 1510, 1510, 171, 1;
1, 420, 8337, 21140, 8337, 420, 1;
1, 988, 40068, 218666, 218666, 40068, 988, 1;
1, 2259, 175296, 1852914, 3935988, 1852914, 175296, 2259, 1;
1, 5065, 717600, 13655760, 55034868, 55034868, 13655760, 717600, 5065, 1;
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(* First program *)
Needs["Combinatorica`"]
T[n_, k_]:= Eulerian[n+1, k]*Binomial[n+1, k]/(k+1);
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* Roger L. Bagula, Apr 14 2010 *)
(* Second program *)
t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k -(m -1))*t[n-1,k,m]];
T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2022 *)
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@CachedFunction
def t(n,k,m):
if (k==1 or k==n): return 1
else: return (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m)
def T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022
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
Showing 1-4 of 4 results.
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