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A274040 Numerator of the rational coefficient at the first power of Pi in Sum_{k>0} (sin(k)/k)^n. 1

%I

%S 1,1,3,1,115,11,43141,733,-722109,-1093,143795597,47489,14249936103,

%T 5276161

%N Numerator of the rational coefficient at the first power of Pi in Sum_{k>0} (sin(k)/k)^n.

%e a(1) = 1, because Sum_{k>0} (sin(k)/k)^1 = (1/2)*Pi - 1/2.

%e a(2) = 1, because Sum_{k>0} (sin(k)/k)^2 = (1/2)*Pi - 1/2.

%e a(3) = 3, because Sum_{k>0} (sin(k)/k)^3 = (3/8)*Pi - 1/2.

%e a(4) = 1, because Sum_{k>0} (sin(k)/k)^4 = (1/3)*Pi - 1/2.

%e This simple pattern breaks starting at n = 7:

%e a(7) = 43141, because Sum_{k>0} (sin(k)/k)^7 = (1/720)*Pi^7 - (7/240)*Pi^6 + (49/192)*Pi^5 - (343/288)*Pi^4 + (2401/768)*Pi^3 - (16807/3840)*Pi^2 + (43141/15360)*Pi - 1/2.

%t a[n_] := Numerator@Coefficient[Sum[Sinc[k]^n, {k, 1, Infinity}], Pi]

%Y Cf. A274041 (denominators).

%K sign,more,hard,frac

%O 1,3

%A _Vladimir Reshetnikov_, Jun 07 2016

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Last modified August 12 07:45 EDT 2022. Contains 356067 sequences. (Running on oeis4.)