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 A026107 Second differences of Motzkin numbers (A001006). 5
 1, 3, 7, 18, 46, 120, 316, 841, 2257, 6103, 16611, 45475, 125139, 345957, 960417, 2676291, 7483299, 20989833, 59042805, 166520124, 470781528, 1333970190, 3787707322, 10775741271, 30711538351, 87677551081, 250704001213, 717923179762 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-1), where T is array in A026105 and U(n,n+1), where U is array in A026120. Also number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 0, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Number of Motzkin paths of length n+1 that start with a (1,1) step and end with a (1,-1) step. - Emeric Deutsch, Jul 11 2001 The sequence 1,1,3,7,18.... has a(n)=sum{k=0..n, C(n,2k)*A000108(k+1) }. - Paul Barry, Jul 18 2003 Equals iterates of M * [1,1,1,1,0,0,0,...] where M = an infinite tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. [From Gary W. Adamson, Jan 08 2009] Motzkin paths of length n-1 that are allowed to go down to the line y=-1 [He-Shapiro, page 38]. - R. J. Mathar, Jul 23 2017 LINKS T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41 FORMULA a(n)=A001006(n+1)-2*A001006(n)+A001006(n-1); g.f.: [(1-z)^2*M-1+z-z^2]/z, where M is the generating function of the Motzkin sequence A001006 (M = 1 + zM + z^2M^2). (n+3)*a(n) +3*(-n-1)*a(n-1) +(-n-3)*a(n-2) +3*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 26 2012 a(n) ~ 2 * 3^(n + 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 17 2019 CROSSREFS Cf. A001006. First differences of A002026. Cf. A026122. Sequence in context: A052960 A059512 A094297 * A173765 A027969 A027971 Adjacent sequences:  A026104 A026105 A026106 * A026108 A026109 A026110 KEYWORD nonn AUTHOR EXTENSIONS Simpler definition from Ralf Stephan, Dec 16 2004 STATUS approved

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Last modified July 12 04:30 EDT 2020. Contains 335658 sequences. (Running on oeis4.)