The Borwein integrals are the class of definite integrals defined by

for odd . The integrals are curious because the terms , 3, ..., 13 all have unit numerators, but , 17, ... do not. 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).

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.

Borwein, D. and Borwein, J. M. "Some Remarkable Properties of Sinc and Related Integrals." Ramanujan J. 5, 73-90, 2001.

Borwein, D.; Borwein, J. M.; and Mares, B. A. Jr. "Multi-Variable Sinc Integrals and Volumes of Polyhedra." Preprint. 2001. http://www.cecm.sfu.ca/preprints/2001pp.html.

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."