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 A203611 Sum_{k=0..n} C(k-1,2*k-1-n)*C(k,2*k-n). 4
 1, 1, 1, 3, 7, 16, 39, 95, 233, 577, 1436, 3590, 9011, 22691, 57299, 145043, 367931, 935078, 2380405, 6068745, 15492702, 39598631, 101323446, 259522398, 665332007, 1707137941, 4383662419, 11264675925, 28966161253, 74530441162, 191879611399, 494265165151 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS For the connection with Fibonacci meanders classified by maximal run length of 1s see the link. Apparently the number of grand Motzkin paths of length n+1 that avoid UU. - David Scambler, Jul 04 2013 LINKS Peter Luschny, Fibonacci meanders. FORMULA For n>0 let A=floor(n/2), R=n-1, B=A-R/2+1, C=A+1, D=A-R and Z=(n+1)/2 if n mod 2 = 1, otherwise Z=n^2*(n+2)/16. Then a(n) = Z*Hypergeometric([1,C,C+1,D,D],[B,B,B-1/2,B+1/2],1/16). G.f.: 2*x/((1+x-x^2)*sqrt((x^2+x+1)*(x^2-3*x+1))-x^4+2*x^3+x^2+2*x-1). - Mark van Hoeij, May 06 2013 a(n) ~ phi^(2*n + 1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 08 2019 MAPLE A203611 := proc(n) local A, R, B, C, D, Z; if n=0 then RETURN(1) fi; A := iquo(n, 2); R := n-1; B := A-R/2+1; C := A+1; D := A-R; Z := `if`(n mod 2 = 1, (n+1)/2, n^2*(n+2)/16); Z*hypergeom([1, C, C+1, D, D], [B, B, B-1/2, B+1/2], 1/16) end: seq(simplify(b(i)), i=0..31); MATHEMATICA a[n_] := Module[{a, r, b, c, d, z}, If[n == 0, Return[1]]; a = Quotient[n, 2]; r = n-1; b = a-r/2+1; c = a+1; d = a-r; z = If[Mod[n, 2] == 1, (n+1)/2, n^2*(n+2)/16]; z*HypergeometricPFQ[{1, c, c+1, d, d}, {b, b, b-1/2, b+1/2}, 1/16] ]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jun 27 2013, translated from Maple *) Table[Sum[Binomial[k-1, 2k-1-n]Binomial[k, 2k-n], {k, 0, n}], {n, 0, 40}] (* Harvey P. Dale, May 25 2014 *) PROG (PARI) x='x+O('x^66); Vec( 2*x/((1+x-x^2) * sqrt((x^2+x+1) * (x^2-3*x+1)) -x^4 +2*x^3 +x^2 +2*x -1) ) \\ Joerg Arndt, May 06 2013 CROSSREFS Cf. A110236, bisection of A202411. Sequence in context: A304937 A152090 A190528 * A176604 A014140 A271788 Adjacent sequences:  A203608 A203609 A203610 * A203612 A203613 A203614 KEYWORD nonn AUTHOR Peter Luschny, Jan 14 2012 STATUS approved

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Last modified February 17 18:13 EST 2020. Contains 332005 sequences. (Running on oeis4.)