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A074092
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Number of plane binary trees of size n+3 and contracted height n.
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3
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1, 2, 8, 40, 144, 448, 1280, 3456, 8960, 22528, 55296, 133120, 315392, 737280, 1703936, 3899392, 8847360, 19922944, 44564480, 99090432, 219152384, 482344960, 1056964608, 2306867200, 5016387584, 10871635968, 23488102400
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OFFSET
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0,2
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LINKS
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FORMULA
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a(0) = 1, a(1) = 2, a(n) = 2^(n-1)*(n+2)*(n-1) = (2^n)*(C(n, n-2)+C(n-1, n-2)) = 2^n * A000096(n-1).
a(n) = 6*a(n-1)-12*a(n-2)+8*a(n-3) for n>4. G.f.: (1-4*x+8*x^2+8*x^3-16*x^4)/(1-2*x)^3. [Colin Barker, Mar 21 2012]
For n>1, a(n) = (1/2) * Sum_{k=0..n+1} Sum_{i=0..n+1} (k-1) * C(n+1,i). - Wesley Ivan Hurt, Sep 20 2017
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MAPLE
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A074092 := n -> `if`((n < 2), n+1, 2^(n-1)*(n+2)*(n-1));
A074092v2 := n -> `if`((n < 2), n+1, (2^n)*(binomial(n, n-2)+binomial(n-1, n-2)));
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MATHEMATICA
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Table[If[n < 2, n + 1, 2^(n - 1)*(n + 2) (n - 1)], {n, 0, 26}] (* or *)
CoefficientList[Series[(1 - 4 x + 8 x^2 + 8 x^3 - 16 x^4)/(1 - 2 x)^3, {x, 0, 26}], x] (* Michael De Vlieger, Sep 22 2017 *)
LinearRecurrence[{6, -12, 8}, {1, 2, 8, 40, 144}, 30] (* Harvey P. Dale, Jun 20 2021 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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