%I A014480
%S A014480 1,6,20,56,144,352,832,1920,4352,9728,21504,47104,102400,221184,475136,
%T A014480 1015808,2162688,4587520,9699328,20447232,42991616,90177536,188743680,
%U A014480 394264576,822083584,1711276032,3556769792,7381975040,15300820992
%N A014480 Expansion of (1+2*x)/(1-2*x)^2.
%C A014480 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.)
%C A014480 Also determinant of the n X n matrix M(i,j)=binomial(2i+2j,i+j) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Mar 27 2004
%C A014480 Subdiagonal in triangle displayed in A128196. - Peter Luschny (peter(AT)luschny.de), 
               Feb 26 2007
%C A014480 Contribution from Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 
               08 2009: (Start)
%C A014480 From two BBP-type formulas by Knuth, (page 6 of the reference)
%C A014480 Sum_{n>=0} 1/a(n) = 2^(1/2)*log(1+2^(1/2))
%C A014480 Sum_{n>=0} (-1)^n/a(n) = 2^(1/2)*atan(1/2^(1/2))
%C A014480 (End)
%H A014480 David Bailey, Peter Borwein, Simon Plouffe, <a href="http://crd.lbl.gov/
               ~dhbailey/dhbpapers/digits.pdf">On the rapid computation of various 
               polylogarithmic constants</a> [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Nov 08 2009]
%F A014480 a(n) = (2n+1)*2^n = 4a(n-1)-4a(n-2) = 4*A052951(n-1) = a(n-1)+A052951(n) 
               = a(n-1)*(2+4/(2n-1)) = A054582(n, n) - Henry Bottomley (se16(AT)btinternet.com), 
               May 16 2001
%F A014480 E.g.f.: x*cosh(sqrt(2)*x) = x + 6x^3/3! + 20x^5/5! + 56x^7/7! +... - 
               Ralf Stephan, Mar 03 2005
%F A014480 a(n)=A118416(n+1,n+1)=A118413(n+1,n+1); A001511(a(n))=A003602(a(n)); 
               A117303(a(n))=a(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 27 2006
%F A014480 Row sums of triangle A132775 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 29 2007
%F A014480 Row sums of triangle A134233 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 14 2007
%F A014480 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 
               2009: (Start)
%F A014480 a(n) = 3*a(n-1) - 2^(n-1)*(2*n-5) with a(0) = 1.
%F A014480 a(n) = 3*a(n-1) - 2*a(n-2) + 2^n with a(0) = 1 and a(1) = 6.
%F A014480 (End)
%e A014480 (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 + ...
%p A014480 a:=n->sum (2^n*n^binomial(j,n)/2,j=1..n): seq(a(n),n=1..29);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2009]
%t A014480 CoefficientList[ Series[(1 + 2*x)/(1 - 2*x)^2, {x, 0, 28}], x]
%Y A014480 Cf. A118417.
%Y A014480 Cf. A128196.
%Y A014480 Cf. A132775.
%Y A014480 Cf. A134233.
%Y A014480 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%Y A014480 Equals the first left hand column of A167580.
%Y A014480 (End)
%Y A014480 Sequence in context: A059822 A152959 A109903 this_sequence A048778 A048611 
               A127982
%Y A014480 Adjacent sequences: A014477 A014478 A014479 this_sequence A014481 A014482 
               A014483
%K A014480 nonn,new
%O A014480 0,2
%A A014480 N. J. A. Sloane (njas(AT)research.att.com).