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A119328
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Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.
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3
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1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 4, -3, 1, 0, 1, -6, 9, -4, 1, 0, -1, 8, -19, 16, -5, 1, 0, 1, -10, 33, -44, 25, -6, 1, 0, -1, 12, -51, 96, -85, 36, -7, 1, 0, 1, -14, 73, -180, 225, -146, 49, -8, 1, 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1
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OFFSET
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0,9
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COMMENTS
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LINKS
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FORMULA
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Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)x^(2j)}
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EXAMPLE
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Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 4, -3, 1,
0, 1, -6, 9, -4, 1,
0, -1, 8, -19, 16, -5, 1,
0, 1, -10, 33, -44, 25, -6, 1,
0, -1, 12, -51, 96, -85, 36, -7, 1,
0, 1, -14, 73, -180, 225, -146, 49, -8, 1,
0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1
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MATHEMATICA
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t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[k, 2 j]*Binomial[i - k, 2 j], {j, 0, i - k}], {i, 0, n}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 25 2013 *)
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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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