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A202411
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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.
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5
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1, 0, 1, 1, 1, 2, 3, 4, 7, 10, 16, 24, 39, 58, 95, 143, 233, 354, 577, 881, 1436, 2204, 3590, 5534, 9011, 13940, 22691, 35213, 57299, 89162, 145043, 226238, 367931, 575114, 935078, 1464382, 2380405, 3734150, 6068745, 9534594, 15492702, 24374230, 39598631
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OFFSET
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0,6
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LINKS
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FORMULA
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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).
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EXAMPLE
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Fibonacci meanders classified by maximal run length of 1s (see the link) lead to the triangle
0, 1;
1, 1, 0, 1;
2, 1, 1, 1, 0, 1;
4, 3, 2, 1, 1, 1, 0, 1;
10, 7, 4, 3, 2, 1, 1, 1, 0, 1;
24, 16, 10, 7, 4, 3, 2, 1, 1, 1, 0, 1.
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MAPLE
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A202411 := proc(n) local A, R, B, C, D, Z, H, J; if n = 0 then RETURN(1) fi;
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:
seq(simplify(A202411(i)), i=0..42);
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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