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Franel Number

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The Franel numbers are the numbers

Fr_n=sum_(k=0)^n(n; k)^3,
(1)

where (n; k) is a binomial coefficient. The first few values for n=0, 1, ... are 1, 2, 10, 56, 346, ... (OEIS A000172). They arise in the first Strehl identity

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

and can be written in closed form as

Fr_n=_3F_2(-n,-n,-n;1,1;-1),
(3)

where _3F_2(a,b,c;d,e;z) is a generalized hypergeometric function.

They are given by the integral

Fr_n=(-1)^nint_0^inftye^(-x)[L_n(x)]^3dx,
(4)

where L_n(x) is a Laguerre polynomial.

They are also given by the recurrence equation

a_n=((7n^2-7n+2)a_(n-1)+8(n-1)^2a_(n-2))/(n^2)
(5)

with a_0=1 and a_1=2.

See also

Apéry's Constant, Binomial Sums, Schmidt's Problem, Strehl Identities

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References

Askey, R. Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM, p. 43, 1975.Barrucand, P. "Problem 75-4: A Combinatorial Identity." SIAM Rev. 17, 168, 1975.Cusick, T. W. "Recurrences for Sums of Powers of Binomial Coefficients." J. Combin. Th. A 52, 77-83, 1989.Franel, J. "On a Question of Laisant." L'intermédiaire des mathématiciens 1, 45-47, 1894.Franel, J. "On a Question of J. Franel." L'intermédiaire des mathématiciens 2, 33-35, 1895.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, p. 193, 1980.Schmidt, A. L. "Generalized q-Legendre Polynomials." J. Comput. Appl. Math. 49, 243-249, 1993.Schmidt, A. L. "Legendre Transforms and Apéry's Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.Sloane, N. J. A. Sequence A000172/M1971 in "The On-Line Encyclopedia of Integer Sequences."

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Franel Number

Cite this as:

Weisstein, Eric W. "Franel Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FranelNumber.html

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