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A099040
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Riordan array (1, 2+2x).
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2
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1, 0, 2, 0, 2, 4, 0, 0, 8, 8, 0, 0, 4, 24, 16, 0, 0, 0, 24, 64, 32, 0, 0, 0, 8, 96, 160, 64, 0, 0, 0, 0, 64, 320, 384, 128, 0, 0, 0, 0, 16, 320, 960, 896, 256, 0, 0, 0, 0, 0, 160, 1280, 2688, 2048, 512, 0, 0, 0, 0, 0, 32, 960, 4480, 7168, 4608, 1024, 0, 0, 0, 0, 0, 0, 384, 4480, 14336, 18432, 10240, 2048
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OFFSET
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0,3
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COMMENTS
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The Riordan array (1,s+t*x) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
T(n,k) is the number of compositions of n into two types of parts of size 1 and 2 that have exactly k parts. - Geoffrey Critzer, Aug 18 2012.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 22 2020
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LINKS
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FORMULA
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Number triangle T(n, k) = 2^k*binomial(k, n-k).
Columns have g.f. (2x+2x^2)^k.
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EXAMPLE
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Rows begin {1}, {0,2}, {0,2,4}, {0,0,8,8}, {0,0,4,24,16}, {0,0,0,24,64,32},...
T(3,2)=8 because we have: 1+2,1+2',1'+2,1'+2',2+1,2+1',2'+1,2'+1' where a part of the second type is designated by '. - Geoffrey Critzer, Aug 18 2012
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MATHEMATICA
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nn = 8; CoefficientList[Series[1/(1 - 2 y x - 2 y x^2), {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 18 2012 *)
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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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