The Franel numbers are the numbers
(1)
where binomial coefficient . The first few
values for A000172 ).
They arise in the first Strehl identity
(2)
and can be written in closed form as
(3)
where generalized hypergeometric
function .
They are given by the integral
(4)
where Laguerre polynomial .
They are also given by the recurrence equation
(5)
with
Sun's conjectures 3.6 and 3.7 (Sun 2022) involve the
weighted binomial transform
(6)
Babanskyy (2026b, 2026c) reported proofs of the corresponding supercongruences modulo prime numbers
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. 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
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. Schmidt,
A. L. "Generalized 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." Referenced
on Wolfram|Alpha Franel Number
Cite this as:
Weisstein, Eric W. "Franel Number." From
MathWorld https://mathworld.wolfram.com/FranelNumber.html
Subject classifications
is a binomial coefficient. The first few
values for 
,
1, ... are 1, 2, 10, 56, 346, ... (OEIS A000172).
They arise in the first Strehl identity
is a generalized hypergeometric
function.
![Fr_n=(-1)^nint_0^inftye^(-x)[L_n(x)]^3dx,](/images/equations/FranelNumber/NumberedEquation4.svg)
is a Laguerre polynomial.
and 
.
for prime numbers 
and truncated sums of 
with denominators 
, 64, and 
. 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.
." 2026c. 
-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.