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 A081294 Expansion of (1-2*x)/(1-4*x). 62
 1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583. Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004 In general, sum{k=0..n, sum{j=0..n, C(2(n-k), j)*C(2k, j)r^j}} has expansion (1-(r+1)x)/(1+(r+3)x+(r-1)(r+3)x^2+(r-1)^3*x^3). - Paul Barry, Jun 04 2005 a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010 Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014 a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016. Index entries for linear recurrences with constant coefficients, signature (4). FORMULA G.f.: (1-2*x)/(1-4*x). a(n) = 4*a(n-1) n>1, with a(0)=1, a(1)=2. a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n>0). E.g.f.: exp(2*x)*cosh(2*x)=(exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003 a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003 a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004 a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005 a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006 a(n) = int(p(n,x)^2/(Pi*sqrt(x(4-x))),x,0,4) where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x)=(2*x-4)*p(n-1,x)-4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007 a(n) = ((2+sqrt(4))^n+(2-sqrt(4))^n)/2. [Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008] a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008 a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009 a(n) = Sum_{k=0..n} A201730(n,k)*3^k . - Philippe Deléham, Dec 06 2011 a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012 G.f.: Q(0), where Q(k)= 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013 E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k)= 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013 a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013 G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017 EXAMPLE G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ... MAPLE a:= n-> 2^max(0, (2*n-1)): seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017 MATHEMATICA CoefficientList[Series[(1-2x)/(1-4x), {x, 0, 40}], x] (* or *) Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *) PROG (PARI) a(n)=1<

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