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Strehl Identities

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The first Strehl identity is the binomial sum identity

sum_(k=0)^n(n; k)^3=sum_(k=0)^n(n; k)^2(2k; n),

(Strehl 1993, 1994; Koepf 1998, p. 55), which are the so-called Franel numbers. For n=1, 2, ..., the first few terms are 1, 2, 10, 56, 346, 2252, 15184, 104960, ... (OEIS A000172).

The second Strehl identity is the binomial sum identity

sum_(k=0)^n(n; k)^2(n+k; k)^2=sum_(k=0)^nsum_(j=0)^n(n; k)(n+k; k)(k; j)^3

(Strehl 1993, 1994; Koepf 1998, p. 55) that is the r=2 case of Schmidt's problem. For n=0, 1, 2, ..., these give the Apéry numbers 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).

See also

Apéry Numbers, Binomial Coefficient, Binomial Sums, Franel Number, Schmidt's Problem

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References

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Schmidt, A. L. "Legendre Transforms and Apéry's Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.Sloane, N. J. A. Sequences A000172/M1971 and A005258/M3057 in "The On-Line Encyclopedia of Integer Sequences."Strehl, V. "Binomial Sums and Identities." Maple Technical Newsletter 10, 37-49, 1993.Strehl, V. "Binomial Identities--Combinatorial and Algorithmic Aspects." Disc. Math. 136, 309-346, 1994.Zudilin, W. "On a Combinatorial Problem of Asmus Schmidt." Elec. J. Combin. 11, R22, 1-8, 2004. https://doi.org/10.37236/1775.

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Strehl Identities

Cite this as:

Weisstein, Eric W. "Strehl Identities." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StrehlIdentities.html

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