The first Strehl identity is the binomial sum identity
(Strehl 1993, 1994; Koepf 1998, p. 55), which are the so-called Franel numbers. For ,
2, ..., the first few terms are 1, 2, 10, 56, 346, 2252, 15184, 104960, ... (OEIS
A000172).
The second Strehl identity is the binomial sum identity
(Strehl 1993, 1994; 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
and A005258/M3057 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." Discrete
Math.136, 309-346, 1994.Zudilin, W. "On a Combinatorial
Problem of Asmus Schmidt." Elec. J. Combin.11, R22, 1-8, 2004.
http://www.combinatorics.org/Volume_11/Abstracts/v11i1r22.html.
,
2, ..., 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).
