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 A099250 Bisection of Motzkin numbers A001006. 5
 1, 4, 21, 127, 835, 5798, 41835, 310572, 2356779, 18199284, 142547559, 1129760415, 9043402501, 73007772802, 593742784829, 4859761676391, 40002464776083, 330931069469828, 2750016719520991, 22944749046030949, 192137918101841817 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. Veronika Irvine, Stephen Melczer, Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018. FORMULA a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^(2*n+1)*sqrt(1-x^2)). [Peter Luschny, Sep 11 2011] Recurrence: (n+1)*(2*n+3)*a(n) = (14*n^2+23*n+6)*a(n-1) + 3*(14*n^2-37*n+21)*a(n-2) - 27*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012 a(n) ~ 3^(2*n+5/2)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012 a(n) = sum(j=0..3*n+4, binomial(j,2*j-5*n-7)*binomial(3*n+4,j))/(3*n+4). [Vladimir Kruchinin, Mar 09 2013] G.f.: (1/x) * Series_Reversion( x*(1+x) / ( (1+2*x)^2 * (1+x+x^2) ) ). - Paul D. Hanna, Oct 03 2014 MAPLE G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G, x=0, 60): seq(coeff(GG, x^(2*n-1)), n=1..24);  # Emeric Deutsch M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end: A099250 := n -> M(2*n+1): seq(A099250(i), i=0..20); # Peter Luschny, Sep 11 2011 MATHEMATICA Take[CoefficientList[Series[(1-x-(1-2x-3x^2)^(1/2))/(2x^2), {x, 0, 60}], x], {2, -1, 2}] (* Harvey P. Dale, Sep 11 2011 *) Table[Hypergeometric2F1[-1/2-n, -n, 2, 4], {n, 0, 30}] (* Jean-François Alcover, Apr 03 2015 *) PROG (PARI) a(n)=sum(j=0, 3*n+4, binomial(j, 2*j-5*n-7)*binomial(3*n+4, j))/(3*n+4); /* Joerg Arndt, Mar 09 2013 */ (PARI) x='x+O('x^66); v=Vec((1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2)); vector(#v\2, n, v[2*n]) \\ Joerg Arndt, May 12 2013 (PARI) {a(n)=polcoeff(1/x*serreverse( x*(1+x)/((1+2*x)^2*(1+x+x^2) +x^2*O(x^n)) ), n)} for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 03 2014 CROSSREFS Cf. A001006, A026945. Sequence in context: A281581 A007345 A255673 * A293192 A300674 A232956 Adjacent sequences:  A099247 A099248 A099249 * A099251 A099252 A099253 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Nov 16 2004 EXTENSIONS More terms from Emeric Deutsch, Nov 17 2004 STATUS approved

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Last modified November 12 12:43 EST 2018. Contains 317109 sequences. (Running on oeis4.)