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A165408
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An aerated Catalan triangle.
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4
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1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 5, 0, 5, 0, 1, 0, 0, 0, 0, 9, 0, 6, 0, 1, 0, 0, 0, 5, 0, 14, 0, 7, 0, 1, 0, 0, 0, 0, 14, 0, 20, 0, 8, 0, 1, 0, 0, 0, 0, 0, 28, 0, 27, 0, 9, 0, 1, 0, 0, 0, 0, 14, 0, 48, 0, 35, 0, 10, 0, 1, 0, 0, 0, 0, 0, 42, 0, 75, 0, 44, 0, 11, 0, 1
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,13
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COMMENTS
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T(n,k) is the number of lattice paths from (0,0) to (k,(n-k)/2) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - Alois P. Heinz, Sep 20 2022
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LINKS
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FORMULA
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T(n,k) = if(n<=3k, C((n+k)/2, k)*((3*k-n)/2 + 1)*(1 + (-1)^(n-k))/(2*(k+1)), 0).
G.f.: 1/(1-x*y-x^3*y/(1-x^3*y/(1-x^3*y/(1-x^3*y/(1-... (continued fraction).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1+(-1)^n)*A001405(n/2)/2.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1+(-1)^n)*A105523(n/2)/2.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A165407(n).
Sum_{k=0..n} 2^k*T(n, k) = A165409(n).
T(2*n, n) = (1+(-1)^n)*A174687(n/2)/2.
T(2*n, n+1) = (1-(-1)^n)*A262394(n/2)/2.
T(2*n, n-1) = (1+(-1)^n)*A236194(n/2)/2
T(3*n-2, n) = A000108(n), n >= 1. (End)
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EXAMPLE
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Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 1, 0, 1;
0, 0, 2, 0, 1;
0, 0, 0, 3, 0, 1;
0, 0, 2, 0, 4, 0, 1;
0, 0, 0, 5, 0, 5, 0, 1;
0, 0, 0, 0, 9, 0, 6, 0, 1;
0, 0, 0, 5, 0, 14, 0, 7, 0, 1;
0, 0, 0, 0, 14, 0, 20, 0, 8, 0, 1;
0, 0, 0, 0, 0, 28, 0, 27, 0, 9, 0, 1;
0, 0, 0, 0, 14, 0, 48, 0, 35, 0, 10, 0, 1;
...
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MAPLE
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b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
end:
T:= (n, k)-> `if`((n-k)::even, b(k, (n-k)/2), 0):
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MATHEMATICA
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b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] + If[y>0, b[x, y-1], 0]], 0];
T[n_, k_]:= If[EvenQ[n-k], b[k, (n-k)/2], 0];
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PROG
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(Magma)
A165408:= func< n, k | n le 3*k select Binomial(Floor((n+k)/2), k)*((3*k-n)/2 +1)*(1+(-1)^(n-k))/(2*(k+1)) else 0 >;
(SageMath)
def A165408(n, k): return 0 if (n>3*k) else binomial(int((n+k)/2), k)*((3*k-n+2)/2 )*(1+(-1)^(n-k))/(2*(k+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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