Given the generating functions defined by

			(1)
			(2)
			(3)

(Sloane's A051028, A051029, and A051030), then

(4)

Hirschhorn (1995) showed that

			(5)
			(6)
			(7)

where

			(8)
			(9)

Hirschhorn (1996) showed that checking the first seven cases to 6 is sufficient to prove the result.

Hirschhorn, M. D. "An Amazing Identity of Ramanujan." Math. Mag. 68, 199-201, 1995.

Hirschhorn, M. D. "A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan." Math. Mag. 69, 267-269, 1996.

Sloane, N. J. A. Sequences A051028, A051029, and A051030 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Ramanujan's Sum Identity." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RamanujansSumIdentity.html