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A124304
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Riordan array (1, x*(1-x^2)).
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2
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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, 1, 0, -4, 0, 1, 0, 0, 0, 3, 0, -5, 0, 1, 0, 0, 0, 0, 6, 0, -6, 0, 1, 0, 0, 0, -1, 0, 10, 0, -7, 0, 1, 0, 0, 0, 0, -4, 0, 15, 0, -8, 0, 1, 0, 0, 0, 0, 0, -10, 0, 21, 0, -9, 0, 1, 0, 0, 0, 0, 1, 0, -20, 0, 28, 0, -10, 0, 1
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OFFSET
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0,13
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COMMENTS
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T(2n,n) is a signed aerated version of C(2n,n).
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2))*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)
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EXAMPLE
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Triangle begins
1;
0, 1;
0, 0, 1;
0, -1, 0, 1;
0, 0, -2, 0, 1;
0, 0, 0, -3, 0, 1;
0, 0, 1, 0, -4, 0, 1;
0, 0, 0, 3, 0, -5, 0, 1;
0, 0, 0, 0, 6, 0, -6, 0, 1;
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MATHEMATICA
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A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
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PROG
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(Magma)
A124304:= func< n, k | (&+[(-1)^j*Binomial(k, k-j)*Binomial(k, n-k-j) : j in [0..n]]) >;
(SageMath)
def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
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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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