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Franel Number

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The Franel numbers are defined for nonnegative integers n by

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 relation

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

for n>=2, with a_n=Fr_n, a_0=1, and a_1=2.

Sun's conjectures 3.6 and 3.7 (Sun 2022) involve the weighted binomial transform

Q_n=sum_(j=0)^n(n; j)(-8)^(n-j)Fr_j.
(6)

Babanskyy (2026bc) reported proofs of the corresponding supercongruences modulo p^2 for prime numbers p and truncated sums of (2n; n)Q_n/t^n with denominators t=-32, 64, and -16. The author reports assistance from OpenAI Codex and GPT-6.1 Sol. As of Oct. 5, 2026, finite checks supported the formulas, but independent specialist review of the complete arguments and a definitive priority check had not been reported.

See also

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

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References

Askey, R. Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM, p. 43, 1975.Babanskyy, O. "Sun's Supercongruence for Two Apéry-Like Binomial Sums." 2026b. https://github.com/aconsciousfractal/Sun-3-6-Supercongruence/blob/bdbea1441a5791058f4662b8959e24feea6f69b7/paper/Sun_3_6_Supercongruence.pdf.Babanskyy, O. "Sun's Supercongruence for Two Apéry-Like Binomial Sums at Discriminant -20." 2026c. https://github.com/aconsciousfractal/Sun-3-7-Supercongruence/blob/4caad01fa469c86a1a95b930293354d843b79d64/paper/Sun_3_7_Supercongruence.pdf.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.Sun, Z.-H. "Supercongruences Involving Apéry-Like Numbers and Binomial Coefficients." AIMS Math. 7, 2729-2781, 2022. https://doi.org/10.3934/math.2022153.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 Resource. https://mathworld.wolfram.com/FranelNumber.html

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