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 A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564. 8
 1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006 Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..2100 D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From N. J. A. Sloane, May 11 2012 Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265. Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3. FORMULA G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002 Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007 a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007 MAPLE seq( sum('binomial(i-2, k)*binomial(i-k, k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001 MATHEMATICA CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]), {x, 0, 30}], x] (* Harvey P. Dale, Jun 12 2016 *) PROG (GAP) List([0..30], i->Sum([0..Int(i/2)], k->Binomial(i-2, k)*Binomial(i-k, k))); # Muniru A Asiru, Mar 09 2019 CROSSREFS First differences of A026135. Row sums of triangle A026105. Pairwise sums of A005727. Cf. A001791, A132816. Cf. A136537. Sequence in context: A279378 A048579 A121449 * A027036 A018040 A018041 Adjacent sequences:  A025563 A025564 A025565 * A025567 A025568 A025569 KEYWORD nonn AUTHOR STATUS approved

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Last modified October 23 12:19 EDT 2019. Contains 328345 sequences. (Running on oeis4.)