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Column 2 of triangle A318945.
4

%I #17 Oct 28 2018 16:44:29

%S 0,0,0,0,1,6,26,97,331,1064,3277,9775,28448,81201,228211,633384,

%T 1740037,4740327,12825008,34500649,92372683,246352952,654878173,

%U 1736172895,4592568896,12125944161,31967715811,84170419272,221388694261,581807602839,1527909651152,4010192518105

%N Column 2 of triangle A318945.

%H Muniru A Asiru, <a href="/A318947/b318947.txt">Table of n, a(n) for n = 0..2000</a>

%H Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L., <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Mathematics (2018), 341(10), 2789-2807.

%F Let alpha(n) = Sum_{k=0..n} binomial(2*n-1-k,k-1)*hypergeom([2,2,1-k], [1,1-2*k+2*n], -1)) then alpha(n) = a(n+3) for n >= 0. - _Peter Luschny_, Oct 28 2018

%F Conjectures from _Colin Barker_, Oct 28 2018: (Start)

%F G.f.: x^4*(1 - x)^3 / ((1 - 2*x)^3*(1 - 3*x + x^2)).

%F a(n) = 9*a(n-1) - 31*a(n-2) + 50*a(n-3) - 36*a(n-4) + 8*a(n-5) for n>7.

%F (End)

%p a := n -> `if`(n < 3, 0, combinat:-fibonacci(2*n) - (n^2 + 9*n + 28)*2^(n - 6)):

%p seq(a(n), n=0..31); # _Peter Luschny_, Oct 28 2018

%o (GAP) Concatenation([0,0,0],List([3..31],n->Fibonacci(2*n)-(n^2+9*n+28)*2^(n-6))); # _Muniru A Asiru_, Oct 28 2018

%Y Cf. A318945.

%K nonn

%O 0,6

%A _N. J. A. Sloane_, Sep 18 2018

%E More terms from _Peter Luschny_, Oct 28 2018

%E a(30) corrected by _Muniru A Asiru_, Oct 28 2018