The Christmas stocking theorem, also known as the hockey stick theorem, states that the sum of a diagonal string of numbers in Pascal's triangle starting at the th entry from the top (where the apex has ) on left edge and continuing down rows is equal to the number to the left and below (the "toe") bottom of the diagonal (the "heel"; Butterworth 2002). This follows from the identity

where is a binomial coefficient.

Butterworth, B. "The Twelve Days of Christmas: Music Meets Math in a Popular Christmas Song." Inside Science News Service, Dec. 17, 2002. http://www.aip.org/isns/reports/2002/058.html.

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Weisstein, Eric W. "Christmas Stocking Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChristmasStockingTheorem.html