OFFSET
1,1
COMMENTS
Sun (2014) conjectures that for any prime p > 3 we have Sum_{k = 0..p-1} M(k)^2 == (2 - 6*p)(p/3) (mod p^2) and Sum_{k = 0..p - 1} k*M(k)^2 == (9*p - 1)(p/3) (mod p^2), where (p/3) is the Legendre symbol.
Sun (2018) proves that a(n) is always an integer.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57(2014), no.7, 1375-1400.
Zhi-Wei Sun, Arithmetic properties of Delannoy numbers and Schroder numbers, J. Number Theory 183(2018), 146-171.
Zhi-Wei Sun, On Motzkin numbers and central trinomial coefficients, arXiv:1801.08905 [math.CO], 2018.
FORMULA
EXAMPLE
a(2) = 23 since (2/2)*Sum_{k=1..2} (2k + 1)*M(k)^2 = (2*1 + 1)*M(1)^2 + (2*2 + 1)*M(2)^2 = 3*1^2 + 5*2^2 = 23.
MAPLE
h := k -> (4*k+2)*hypergeom([(1-k)/2, -k/2], [2], 4)^2:
a := proc(n) add(simplify(h(k)), k=1..n): if % mod n = 0 then %/n else -1 fi end:
seq(a(n), n=1..25); # Peter Luschny, Nov 16 2017
MATHEMATICA
M[n_] := M[n] = Sum[Binomial[n, 2k] Binomial[2k, k]/(k + 1), {k, 0, n/2}];
a[n_] := a[n] = 2/n * Sum[(2k + 1) M[k]^2, {k, 1, n}];
Table[a[n], {n, 1, 25}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 15 2017
STATUS
approved