(Strehl 1993; Strehl 1994, p. 327; Koepf 1998, p. 55), whose values are the so-called Franel numbers. For , 1, ..., the first few terms are 1, 2, 10, 56, 346, 2252,
15184, 104960, ... (OEIS A000172).
The second is
(Strehl 1993; Strehl 1994, p. 309; Koepf 1998, p. 55) that is the case of Schmidt's
problem. For ,
1, 2, ..., these give the Apéry numbers 1,
5, 73, 1445, 33001, 819005, ... (OEIS A005259).
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. A58,
358-375, 1995.Sloane, N. J. A. Sequences A000172/M1971,
A005258/M3057, and A005259
in "The On-Line Encyclopedia of Integer Sequences."Strehl,
V. "Binomial Sums and Identities." Maple Technical Newsletter10,
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,
No. 1, R22, 2004. https://doi.org/10.37236/1775.
. The first is
, 1, ..., the first few terms are 1, 2, 10, 56, 346, 2252,
15184, 104960, ... (OEIS A000172).
case of Schmidt's
problem. For 
,
1, 2, ..., these give the Apéry numbers 1,
5, 73, 1445, 33001, 819005, ... (OEIS A005259).
