A081294 Expansion of (1-2*x)/(1-4*x).

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Author

Paul Barry, Mar 17 2003

Keywords

• easy
• nonn

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(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]

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

INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021

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

Examples

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...
		

References

• Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

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 to divisibility sequences.
• Index entries for linear recurrences with constant coefficients, signature (4).

Crossrefs

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

Programs

Magma

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

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 *)

PARI

a(n)=1<Charles R Greathouse IV, Jul 25 2011

PARI

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

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) = 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

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

Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022

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