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

1, 2, 6, 32, 230, 2112, 23548, 309248, 4675014, 79969280, 1527092468, 32203259904, 743288515164, 18638209056768, 504541774904760, 14664951970922496, 455522635895576646, 15058911973677465600, 527896878148304296900, 19559986314930028544000, 763820398700983273655796, 31353195811771939838492672

Offset: 1

Author

N. J. A. Sloane, Aug 18 2015

Keywords

• nonn

Links

• 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.

Crossrefs

Cf. A049330, A049331, A099765.

Programs

Magma

[(&+[(-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

Maple

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

Mathematica

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

PARI

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

Sage

[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

Formula

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