Klee's Identity

Klee's identity is the binomial sum

where is a binomial coefficient. For , 1, ... and , 1,..., the following array is obtained.

1	0	0	0	0	0	0	0	0	0	0
0		0	0	0	0	0	0	0	0	0
0		1	0	0	0	0	0	0	0	0
0	0	2		0	0	0	0	0	0	0
0	0	1		1	0	0	0	0	0	0
0	0	0		4		0	0	0	0	0
0	0	0		6		1	0	0	0	0
0	0	0	0	4		6		0	0	0
0	0	0	0	1		15		1	0	0
0	0	0	0	0		20		8		0
0	0	0	0	0		15		28		1

(OEIS A092865)

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More things to try:

binomial coefficients
Abel's binomial theorem
{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}

References

Riordan, J. Combinatorial Identities. New York: Wiley, p. 13, 1979.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.Sloane, N. J. A. Sequence A092865 in "The On-Line Encyclopedia of Integer Sequences."

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Klee's Identity

Cite this as:

Weisstein, Eric W. "Klee's Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KleesIdentity.html

Klee's Identity

See also

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References

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Subject classifications

1	0	0	0	0	0	0	0	0	0	0
0		0	0	0	0	0	0	0	0	0
0		1	0	0	0	0	0	0	0	0
0	0	2		0	0	0	0	0	0	0
0	0	1		1	0	0	0	0	0	0
0	0	0		4		0	0	0	0	0
0	0	0		6		1	0	0	0	0
0	0	0	0	4		6		0	0	0
0	0	0	0	1		15		1	0	0
0	0	0	0	0		20		8		0
0	0	0	0	0		15		28		1

1	0	0	0	0	0	0	0	0	0	0
0		0	0	0	0	0	0	0	0	0
0		1	0	0	0	0	0	0	0	0
0	0	2		0	0	0	0	0	0	0
0	0	1		1	0	0	0	0	0	0
0	0	0		4		0	0	0	0	0
0	0	0		6		1	0	0	0	0
0	0	0	0	4		6		0	0	0
0	0	0	0	1		15		1	0	0
0	0	0	0	0		20		8		0
0	0	0	0	0		15		28		1

1	0	0	0	0	0	0	0	0	0	0
0		0	0	0	0	0	0	0	0	0
0		1	0	0	0	0	0	0	0	0
0	0	2		0	0	0	0	0	0	0
0	0	1		1	0	0	0	0	0	0
0	0	0		4		0	0	0	0	0
0	0	0		6		1	0	0	0	0
0	0	0	0	4		6		0	0	0
0	0	0	0	1		15		1	0	0
0	0	0	0	0		20		8		0
0	0	0	0	0		15		28		1