A290265 - OEIS

%I #13 Aug 04 2017 20:02:07

%S 0,0,0,2,4,16,38,114,288,800,2092,5702,15244,41568,112884,309822,

%T 851344,2354656,6530336,18193238,50834716,142530256,400713502,

%U 1129710694,3192584432,9043259136,25669403892,73007358218,208022076292,593741582912,1697381979094,4859758184274,13933559180928

%N The number of non-palindromic Motzkin paths of length n.

%C The Motzkin paths (A001006) are classified here as either palindromic or non-palindromic. The latter are counted by the sequence 1, 1, 2, 2, 3, 3, 5, 5..., offset 0, i.e., entries of A005773 repeated.

%C Non-palindromic means, there is at least one step of the n (say, the s-th) which does not match the (n-s+1)st step. Not matching means, if the s-th step is U, the (n-s+1)st step is not D, or if the s-th step is F (sometimes also denoted H), the (n-s+1)st step is not F.

%C All terms are even (because a non-palindromic path reversed creates a different non-palindromic path).

%F a(n) = A001006(n)-A005773(1+floor(n/2)).

%F Conjecture: -(16*n-47)*(n+2)*(n+1)*a(n) -(n+1)*(9*n^2-167*n+188)*a(n-1) +n*(139*n^2-450*n+59)*a(n-2) +(n-1)*(187*n^2-1619*n+2250)*a(n-3) -(n-2)*(97*n^2+346*n-2255)*a(n-4) +(-311*n^3+2398*n^2-5779*n+4112)*a(n-5) +3*(-153*n^3+1458*n^2-4361*n+4348)*a(n-6) -3*(n-6)*(169*n^2-1152*n+1799)*a(n-7) -9*(n-6)*(n-7)*(23*n-82)*a(n-8)=0.

%p A290265 := proc(n)

%p     A001006(n)-A005773(1+floor(n/2)) ;

%p end proc:

%p seq( A290265(n),n=0..60) ;

%t a001006[n_]:=Hypergeometric2F1[(1-n)/2, -n/2, 2, 4]; a005773[n_]:=If[n==0, 1, Sum[k*Sum[Binomial[n, j]*Binomial[j, 2*j-n-k], {j, 0, n}]/n, {k, 1, n}]]; Table[a001006[n] - a005773[1 + Floor[n/2]], {n, 0, 50}] (* _Indranil Ghosh_, Aug 04 2017 *)

%Y Cf. A001006, A005773.

%K nonn

%O 0,4

%A _R. J. Mathar_, Jul 25 2017