cp's OEIS Frontend

A202411 a(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.

%I A202411 #16 Mar 16 2023 06:18:08
%S A202411 1,0,1,1,1,2,3,4,7,10,16,24,39,58,95,143,233,354,577,881,1436,2204,
%T A202411 3590,5534,9011,13940,22691,35213,57299,89162,145043,226238,367931,
%U A202411 575114,935078,1464382,2380405,3734150,6068745,9534594,15492702,24374230,39598631
%N A202411 a(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 A202411 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FibonacciMeanders">Fibonacci meanders</a>.
%F A202411 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:
%F A202411 a(n) = Z*Hypergeometric([1, C, C+1, D, D-J], [B, B, B-1/2, B+1/2-J], 1/16).
%e A202411 Fibonacci meanders classified by maximal run length of 1s (see the link) lead to the triangle
%e A202411    0,  1;
%e A202411    1,  1,  0, 1;
%e A202411    2,  1,  1, 1, 0, 1;
%e A202411    4,  3,  2, 1, 1, 1, 0, 1;
%e A202411   10,  7,  4, 3, 2, 1, 1, 1, 0, 1;
%e A202411   24, 16, 10, 7, 4, 3, 2, 1, 1, 1, 0, 1.
%p A202411 A202411 := proc(n) local A, R, B, C, D, Z, H, J; if n = 0 then RETURN(1) fi;
%p A202411 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 A202411 seq(simplify(A202411(i)),i=0..42);
%t A202411 A202411[0] = 1; A202411[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[A202411[n], {n, 0, 42}]
%t A202411 (* _Jean-François Alcover_, Jan 27 2014, translated from Maple *)
%Y A202411 Cf. A110236, A203611.
%K A202411 nonn
%O A202411 0,6
%A A202411 _Peter Luschny_, Jan 14 2012