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

1, 1, 3, 1, 115, 11, 43141, 733, -722109, -1093, 143795597, 47489, 14249936103, 5276161

Offset

1,3

Links

Table of n, a(n) for n=1..14.

Example

a(1) = 1, because Sum_{k>0} (sin(k)/k)^1 = (1/2)*Pi - 1/2.
a(2) = 1, because Sum_{k>0} (sin(k)/k)^2 = (1/2)*Pi - 1/2.
a(3) = 3, because Sum_{k>0} (sin(k)/k)^3 = (3/8)*Pi - 1/2.
a(4) = 1, because Sum_{k>0} (sin(k)/k)^4 = (1/3)*Pi - 1/2.
This simple pattern breaks starting at n = 7:
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.

Mathematica

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

Crossrefs

Cf. A274041 (denominators).

Sequence in context: A221195 A071291 A049330 * A367948 A369187 A266363

Adjacent sequences: A274037 A274038 A274039 * A274041 A274042 A274043

Keyword

sign,more,hard,frac

Author

Vladimir Reshetnikov, Jun 07 2016

Status

approved