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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 August 10 04:12 EDT 2024. Contains 375044 sequences. (Running on oeis4.)