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

%I A261398 #28 Jun 25 2022 10:01:04
%S A261398 1,2,6,32,230,2112,23548,309248,4675014,79969280,1527092468,
%T A261398 32203259904,743288515164,18638209056768,504541774904760,
%U A261398 14664951970922496,455522635895576646,15058911973677465600,527896878148304296900,19559986314930028544000,763820398700983273655796,31353195811771939838492672
%N A261398 Integer coefficients arising from a formula for Sum_{m>=1} sin(Pi*m/3)^2/m^2.
%H A261398 Vincenzo Librandi, <a href="/A261398/b261398.txt">Table of n, a(n) for n = 1..200</a>
%H A261398 R. Butler, <a href="https://www.jstor.org/stable/2309178">On the Evaluation of Integral_{x=0..oo} (sin(t))^m/t^m dt by the Trapezoidal Rule</a>, The American Mathematical Monthly, vol. 67, no. 6, 1960, pp. 566-69.
%H A261398 J. W. H. Swanepoel, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.002">On a generalization of a theorem by Euler</a>, Journal of Number Theory 149 (2015) 46-56.
%F A261398 a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i*binomial(n,i)*(n-2*i)^(n-1).
%F A261398 a(n)/(2^n*(n-1)!) = A049330(n)/A049331(n).
%F A261398 a(n) = n * A099765(n). - _Vladimir Reshetnikov_, Sep 05 2016
%p A261398 A261398 := proc(n)
%p A261398     add( (-1)^i*binomial(n,i)*(n-2*i)^(n-1),i=0..floor((n-1)/2)) ;
%p A261398 end proc:
%p A261398 seq(A261398(n),n=1..25) ; # _R. J. Mathar_, Aug 19 2015
%t A261398 Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}], {n, 1, 20}] (* _Vladimir Reshetnikov_, Sep 05 2016 *)
%o A261398 (PARI) a(n) = sum(i=0, (n-1)\2, (-1)^i*binomial(n,i)*(n-2*i)^(n-1)); \\ _Michel Marcus_, Sep 05 2016
%o A261398 (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
%o A261398 (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
%Y A261398 Cf. A049330, A049331, A099765.
%K A261398 nonn
%O A261398 1,2
%A A261398 _N. J. A. Sloane_, Aug 18 2015