for positive odd integers. If , with for and , then , where is a nonnegative integer
and
denotes the double factorial (Borwein and Borwein
2001, pp. 74, 78, 86). These sinc function
integrals satisfy
for ,
1, ..., 6, but .
Thus the values
for ,
3, ..., 13 have unit numerators, and this pattern first
fails at .
The sequence of values of for , 3, ... is given by 1/2, 1/6, 1/30, 1/210, 1/1890, 1/20790,
1/270270, 467807924713440738696537864469/1896516717212415135141110350293750000, ...
(OEIS A068214 and A068215;
Borwein et al. 2004, p. 98; Bailey et al. 2006). Both OEIS sequences
use index ,
starting at 0. Sequence OEIS A068215 gives
the denominators of . Sequence OEIS A068214
gives the numerators of , which agree with the numerators
of the explicitly displayed values of .
Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer.
Math. Monthly113, 481-509, 2006.Borwein, D. and Borwein,
J. M. "Some Remarkable Properties of Sinc and Related Integrals."
Ramanujan J.5, 73-89, 2001. https://doi.org/10.1023/A:1011497229317.Borwein,
D.; Borwein, J. M.; and Mares, B. A. Jr. "Multi-Variable Sinc Integrals
and Volumes of Polyhedra." Ramanujan J.6, 189-208, 2002.Borwein,
J.; Bailey, D.; and Girgensohn, R. "Some Curious Sinc Integrals." §2.5
in Experimentation
in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters,
p. 98, 2004.Sloane, N. J. A. Sequences A068214
and A068215 in "The On-Line Encyclopedia
of Integer Sequences."
. If 
, with 
for 
and 
, then ![I_(2m+1)=J_m/[pi(2m+1)!!]](/images/equations/BorweinIntegrals/Inline6.svg)
, where 
is a nonnegative integer
and 
denotes the double factorial (Borwein and Borwein
2001, pp. 74, 78, 86). These sinc function
integrals satisfy 
for 
,
1, ..., 6, but 
.
Thus the values 
for 
,
3, ..., 13 have unit numerators, and this pattern first
fails at 
.
The sequence of values of 
for 
, 3, ... is given by 1/2, 1/6, 1/30, 1/210, 1/1890, 1/20790,
1/270270, 467807924713440738696537864469/1896516717212415135141110350293750000, ...
(OEIS A068214 and A068215;
Borwein et al. 2004, p. 98; Bailey et al. 2006). Both OEIS sequences
use index 
,
starting at 0. Sequence OEIS A068215 gives
the denominators of 
. Sequence OEIS A068214
gives the numerators of 
, which agree with the numerators
of the explicitly displayed values of 
.
