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A290265 The number of non-palindromic Motzkin paths of length n. 1
0, 0, 0, 2, 4, 16, 38, 114, 288, 800, 2092, 5702, 15244, 41568, 112884, 309822, 851344, 2354656, 6530336, 18193238, 50834716, 142530256, 400713502, 1129710694, 3192584432, 9043259136, 25669403892, 73007358218, 208022076292, 593741582912, 1697381979094, 4859758184274, 13933559180928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

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.

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

LINKS

Table of n, a(n) for n=0..32.

FORMULA

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

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.

MAPLE

A290265 := proc(n)

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

end proc:

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A001006, A005773.

Sequence in context: A110998 A051861 A296297 * A223093 A333022 A048222

Adjacent sequences:  A290262 A290263 A290264 * A290266 A290267 A290268

KEYWORD

nonn

AUTHOR

R. J. Mathar, Jul 25 2017

STATUS

approved

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Last modified June 22 01:32 EDT 2021. Contains 345367 sequences. (Running on oeis4.)