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A295132 a(n) = (2/n)*Sum_{k=1..n} (2k+1)*M(k)^2 where M(k) is the Motzkin number A001006(k).

%I A295132 #39 Nov 11 2022 03:30:49
%S A295132 6,23,90,432,2286,13176,80418,513764,3400518,23167311,161640554,
%T A295132 1150633512,8332048638,61232315553,455830692210,3432015694314,
%U A295132 26101221114582,200295455169015,1549473966622602,12074304397434552,94713783502786686,747454269790900728
%N A295132 a(n) = (2/n)*Sum_{k=1..n} (2k+1)*M(k)^2 where M(k) is the Motzkin number A001006(k).
%C A295132 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.
%C A295132 Sun (2018) proves that a(n) is always an integer.
%H A295132 Zhi-Wei Sun, <a href="/A295132/b295132.txt">Table of n, a(n) for n = 1..200</a>
%H A295132 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11425-014-4809-z">Congruences involving generalized central trinomial coefficients</a>, Sci. China Math. 57(2014), no.7, 1375-1400.
%H A295132 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2017.07.011">Arithmetic properties of Delannoy numbers and Schroder numbers</a>, J. Number Theory 183(2018), 146-171.
%H A295132 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1801.08905">On Motzkin numbers and central trinomial coefficients</a>, arXiv:1801.08905 [math.CO], 2018.
%F A295132 a(n) = 2*A005043(n+1)*((6+6/n)*A005043(n) + (2+1/n)*A005043(n+1)). - _Mark van Hoeij_, Nov 10 2022
%e A295132 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.
%p A295132 h := k -> (4*k+2)*hypergeom([(1-k)/2,-k/2],[2],4)^2:
%p A295132 a := proc(n) add(simplify(h(k)),k=1..n): if % mod n = 0 then %/n else -1 fi end:
%p A295132 seq(a(n), n=1..25); # _Peter Luschny_, Nov 16 2017
%t A295132 M[n_] := M[n] = Sum[Binomial[n, 2k] Binomial[2k, k]/(k + 1), {k, 0, n/2}];
%t A295132 a[n_] := a[n] = 2/n * Sum[(2k + 1) M[k]^2, {k, 1, n}];
%t A295132 Table[a[n], {n, 1, 25}]
%Y A295132 Cf. A001006, A005043, A295112, A295113.
%K A295132 nonn
%O A295132 1,1
%A A295132 _Zhi-Wei Sun_, Nov 15 2017