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A060920
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Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
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10
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1, 2, 1, 5, 5, 1, 13, 20, 9, 1, 34, 71, 51, 14, 1, 89, 235, 233, 105, 20, 1, 233, 744, 942, 594, 190, 27, 1, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1, 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1
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OFFSET
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0,2
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COMMENTS
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Companion triangle (odd-indexed members) A060921.
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LINKS
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FORMULA
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T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.
T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).
Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).
G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).
G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003
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EXAMPLE
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Triangle begins as:
1;
2, 1;
5, 5, 1;
13, 20, 9, 1;
34, 71, 51, 14, 1;
89, 235, 233, 105, 20, 1;
233, 744, 942, 594, 190, 27, 1;
610, 2285, 3522, 2860, 1295, 315, 35, 1;
1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1;
4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;
10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;
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MATHEMATICA
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A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j, 0, 2*n-k}];
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PROG
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(Magma)
A060920:= func< n, k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
(Sage)
def A060920(n, k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
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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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