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A202411 Sum_{k=floor(n/4)..R} C(k,m*k-(-1)^n*(R-k))*C(k+1,m*(k+2)-(-1)^n*(R-k+1)) where m = (n+1) mod 2 and R = (n+m-3)/2 for n>0 and a(0)=1. 5

%I

%S 1,0,1,1,1,2,3,4,7,10,16,24,39,58,95,143,233,354,577,881,1436,2204,

%T 3590,5534,9011,13940,22691,35213,57299,89162,145043,226238,367931,

%U 575114,935078,1464382,2380405,3734150,6068745,9534594,15492702,24374230,39598631

%N Sum_{k=floor(n/4)..R} C(k,m*k-(-1)^n*(R-k))*C(k+1,m*(k+2)-(-1)^n*(R-k+1)) where m = (n+1) mod 2 and R = (n+m-3)/2 for n>0 and a(0)=1.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FibonacciMeanders">Fibonacci meanders</a>.

%F For n>0 let H=floor(n/2), A=floor(H/2), R=H-1, B=A-R/2+1, C=A+1, D=A-R, J=n mod 2 and Z = if(H mod 2 = 1,(H+1)/2,H^2*(H+2)/16) if J = 0 else Z = if(H mod 2 = 1,1, H*(H+2)/4). Then a(n) = Z*Hypergeometric([1,C,C+1,D,D-J],[B,B,B-1/2,B+1/2-J],1/16).

%e Fibonacci meanders classified by maximal run length of 1s (see the link) lead to the triangle

%e 0,1;

%e 1,1,0,1;

%e 2,1,1,1,0,1;

%e 4,3,2,1,1,1,0,1;

%e 10,7,4,3,2,1,1,1,0,1;

%e 24,16,10,7,4,3,2,1,1,1,0,1.

%p A202411 := proc(n) local A, R, B, C, D, Z, H, J; if n = 0 then RETURN(1) fi;

%p H:=iquo(n,2); A:=iquo(H,2); R:=H-1; B:=A-R/2+1; C:=A+1; D:=A-R; J:=n mod 2; if J = 0 then Z:=`if`(H mod 2 = 1,(H+1)/2,H^2*(H+2)/16) else Z:=`if`(H mod 2 = 1,1, H*(H+2)/4) fi; Z*hypergeom([1,C,C+1,D,D-J],[B,B,B-1/2,B+1/2-J],1/16) end:

%p seq(simplify(A202411(i)),i=0..42);

%t a[0] = 1; a[n_] := Module[{A, R, B, C, D, Z, H, J}, H = Quotient[n, 2]; A = Quotient[H, 2]; R = H-1; B = A-R/2+1; C = A+1; D = A-R; J = Mod[n, 2]; If[J == 0, Z = If[Mod[H, 2] == 1, (H+1)/2, H^2*(H+2)/16], Z = If[Mod[H, 2] == 1, 1, H*(H+2)/4]]; Z*HypergeometricPFQ[{1, C, C+1, D, D-J}, {B, B, B-1/2, B+1/2-J}, 1/16]]; Table[a[n], {n, 0, 42}] (* _Jean-Fran├žois Alcover_, Jan 27 2014, translated from Maple *)

%Y Cf. A110236, A203611.

%K nonn

%O 0,6

%A _Peter Luschny_, Jan 14 2012

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Last modified February 24 12:47 EST 2020. Contains 332209 sequences. (Running on oeis4.)