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A014480
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Expansion of (1+2*x)/(1-2*x)^2.
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42
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1, 6, 20, 56, 144, 352, 832, 1920, 4352, 9728, 21504, 47104, 102400, 221184, 475136, 1015808, 2162688, 4587520, 9699328, 20447232, 42991616, 90177536, 188743680, 394264576, 822083584, 1711276032, 3556769792, 7381975040, 15300820992
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OFFSET
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0,2
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COMMENTS
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Number of binary trees of size n and height n-1, computed from size n=3 onward; i.e. A014480(n) = A073345(n+3,n+2). (For sizes n=0 through 2 there are no such trees.)
Also determinant of the n X n matrix M(i,j)=binomial(2i+2j,i+j). - Benoit Cloitre, Mar 27 2004
From two BBP-type formulas by Knuth, (page 6 of the reference)
Sum_{n>=0} 1/a(n) = 2^(1/2)*log(1+2^(1/2))
Sum_{n>=0} (-1)^n/a(n) = 2^(1/2)*atan(1/2^(1/2))
(End)
Create a triangle with first column T(n,1)=1+4*n for n=0 1 2... The remaining terms T(r,c)=T(r,c-1)+T(r-1,c-1). T(n,n+1))=a(n). - J. M. Bergot, Dec 18 2012
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LINKS
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FORMULA
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E.g.f.: x*cosh(sqrt(2)*x) = x + 6x^3/3! + 20x^5/5! + 56x^7/7! +... - Ralf Stephan, Mar 03 2005
a(n) = 3*a(n-1) - 2^(n-1)*(2*n-5) with a(0) = 1.
a(n) = 3*a(n-1) - 2*a(n-2) + 2^n with a(0) = 1 and a(1) = 6.
(End)
G.f.: -G(0) where G(k) = 1 - (2*k+2)/(1 - x/(x - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
E.g.f.: Q(0), where Q(k)= 1 + 4*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
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EXAMPLE
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(1 + 2*x)/(1-2*x)^2 = 1 + 6*x + 20*x^2 + 56*x^3 + 144*x^4 + 352*x^5 + 832*x^6 + ...
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MAPLE
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a:=n-> sum(2^n*n^binomial(j, n)/2, j=1..n): seq(a(n), n=1..29); # Zerinvary Lajos, Apr 18 2009
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MATHEMATICA
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CoefficientList[ Series[(1 + 2*x)/(1 - 2*x)^2, {x, 0, 28}], x]
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PROG
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(Haskell)
a014480 n = a014480_list !! n
a014480_list = 1 : 6 : map (* 4)
(zipWith (-) (tail a014480_list) a014480_list)
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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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