Franel Number
The Franel numbers are the numbers
 |
(1)
|
where 
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
 |
(2)
|
and can be written in closed form as
 |
(3)
|
where 
is a generalized
hypergeometric function.
They are given by the integral
![Fr_n=(-1)^nint_0^inftye^(-x)[L_n(x)]^3dx,](/images/equations/FranelNumber/NumberedEquation4.gif) |
(4)
|
where 
is a Laguerre
polynomial.
They are also given by the recurrence equation
 |
(5)
|
with 
and 
.
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 
-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."
Weisstein, Eric W. "Franel Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FranelNumber.html
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Abel's binomial theorem