A002297 Numerator of (2/Pi)*Integral_{0..inf} (sin x / x)^n dx. (Formerly M2262 N0893)

1, 1, 3, 2, 115, 11, 5887, 151, 259723, 15619, 381773117, 655177, 20646903199, 27085381, 467168310097, 2330931341, 75920439315929441, 12157712239, 5278968781483042969, 37307713155613, 9093099984535515162569, 339781108897078469, 168702835448329388944396777

Offset

1,3

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..100

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.

Formula

a(n) = numerator((n/2^(n-1)) * sum((-1)^r*(n-2*r)^(n-1)/(r!*(n-r)!), r=0..n/2)). - Sean A. Irvine, Oct 01 2013

Example

1, 1, 3/4, 2/3, 115/192, 11/20, ...

Mathematica

a[n_] := Numerator[ (2/Pi)*Integrate[ (Sin[x]/x)^n, {x, 0, Infinity}] ]; Table[ a[n], {n, 1, 21}] (* Jean-François Alcover, Dec 19 2011 *)
Numerator@Table[Sum[(-1)^k (n-2k)^(n-1) Binomial[n, k], {k, 0, n/2}]/((n-1)! 2^(n-1)), {n, 1, 30}] (* Vladimir Reshetnikov, Sep 02 2016 *)

Prog

(PARI) a(n) = numerator((n/2^(n-1)) * sum(r=0, n/2, (-1)^r*(n-2*r)^(n-1)/(r!*(n-r)!))); \\ Michel Marcus, Oct 02 2013

Crossrefs

Cf. A002298 (for denominators), A002304, A002305. Essentially the same as A049330, except for the n=4 term.

Sequence in context: A358596 A323745 A109899 * A183270 A152017 A076931

Adjacent sequences: A002294 A002295 A002296 * A002298 A002299 A002300

Keyword

nonn,frac,easy,nice

Author

N. J. A. Sloane

Extensions

a(22)-a(23) from T. D. Noe, Feb 22 2014

Status

approved