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A178112
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Number triangle T(n,k)=C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
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3
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1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1
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OFFSET
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0,13
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COMMENTS
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Coefficient array of polynomials P(n,x)=xP(n-1,x)+((1+(-1)^n)/2)*P(n-2,x), P(0,x)=1,P(1,x)=x.
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LINKS
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EXAMPLE
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Triangle begins
1,
0, 1,
1, 0, 1,
0, 1, 0, 1,
1, 0, 2, 0, 1,
0, 1, 0, 2, 0, 1,
1, 0, 3, 0, 3, 0, 1,
0, 1, 0, 3, 0, 3, 0, 1,
1, 0, 4, 0, 6, 0, 4, 0, 1,
0, 1, 0, 4, 0, 6, 0, 4, 0, 1,
1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1
Production matrix is
0, 1,
1, 0, 1,
0, 0, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Production matrix of inverse is
0, 1,
-1, 0, 1,
0, 0, 0, 1,
0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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MAPLE
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binomial(floor(n/2), floor(k/2))*( 1+(-1)^(n-k) )/2 ;
end proc:
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MATHEMATICA
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Table[Binomial[Floor[n/2], Floor[k/2]]*(1 + (-1)^(n - k))/2, {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 31 2020 *)
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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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