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%I A081294 #118 Jan 20 2025 03:57:07
%S A081294 1,2,8,32,128,512,2048,8192,32768,131072,524288,2097152,8388608,
%T A081294 33554432,134217728,536870912,2147483648,8589934592,34359738368,
%U A081294 137438953472,549755813888,2199023255552,8796093022208,35184372088832
%N A081294 Expansion of (1-2*x)/(1-4*x).
%C A081294 Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
%C A081294 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
%C A081294 In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, 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 [corrected by _Jason Yuen_, Jan 20 2025]
%C A081294 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
%C A081294 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
%C A081294 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
%C A081294 INVERT transform of powers of 2 (A000079). - _Alois P. Heinz_, Feb 11 2021
%C A081294 a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - _Geoffrey Critzer_, Apr 21 2023
%D A081294 Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.
%H A081294 Vincenzo Librandi, <a href="/A081294/b081294.txt">Table of n, a(n) for n = 0..1000</a>
%H A081294 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A081294 M. Paukner, L. Pepin, M. Riehl, and J. Wieser, <a href="https://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.
%H A081294 <a href="/index/Di#divseq">Index to divisibility sequences</a>.
%H A081294 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (4).
%F A081294 G.f.: (1-2*x)/(1-4*x).
%F A081294 a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
%F A081294 a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
%F A081294 E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - _Paul Barry_, May 10 2003
%F A081294 a(n) = Sum_{k=0..n} C(2*n, 2*k). - _Paul Barry_, May 20 2003
%F A081294 a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - _Paul Barry_, Mar 10 2004
%F A081294 a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - _Paul Barry_, Jan 12 2005
%F A081294 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - _Paul Barry_, Jun 04 2005
%F A081294 a(n) = Sum_{k=0..n} A038763(n,k). - _Philippe Deléham_, Sep 22 2006
%F A081294 a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, 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
%F A081294 a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
%F A081294 a(n) = A000079(n) * A011782(n). - _Philippe Deléham_, Dec 01 2008
%F A081294 a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - _Jaroslav Krizek_, Jul 27 2009
%F A081294 a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - _Philippe Deléham_, Dec 06 2011
%F A081294 a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - _Philippe Deléham_, Feb 04 2012
%F A081294 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
%F A081294 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
%F A081294 a(n) = ceiling( 2^(2n-1) ). - _Wesley Ivan Hurt_, Jun 30 2013
%F A081294 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
%F A081294 Sum_{n>=0} 1/a(n) = 5/3. - _Amiram Eldar_, Aug 18 2022
%F A081294 Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - _Geoffrey Critzer_, Apr 21 2023
%e A081294 G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...
%p A081294 a:= n-> 2^max(0, (2*n-1)):
%p A081294 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 20 2017
%t A081294 CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
%t A081294 Join[{1}, NestList[4 # &, 2, 40]] (* _Harvey P. Dale_, Apr 22 2011 *)
%o A081294 (PARI) a(n)=1<<max(0,2*n-1) \\ _Charles R Greathouse IV_, Jul 25 2011
%o A081294 (Magma) [(4^n+0^n)/2: n in [0..30]]; // _Vincenzo Librandi_, Jul 26 2011
%o A081294 (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // _Marius A. Burtea_, Jan 20 2020
%o A081294 (PARI) x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ _Altug Alkan_, Dec 21 2015
%Y A081294 Row sums of triangle A136158.
%Y A081294 Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.
%K A081294 easy,nonn
%O A081294 0,2
%A A081294 _Paul Barry_, Mar 17 2003

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