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A081294
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Expansion of (1-2*x)/(1-4*x).
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41
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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; internal format)
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OFFSET
| 0,2
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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 (pbarry(AT)wit.ie), 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) [From Toby Gottfried, Mar 22 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index to sequences with linear recurrences with constant coefficients, signature (4).
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FORMULA
| G.f.: (1-2*x)/(1-4*x).
a(0)=1, a(1)=2, a(n)=4*a(n-1), n>1.
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<=k<=n} A038763(n,k) . - Philippe DELEHAM, 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. [From Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008]
a(n) = A000079(n)*A011782(n). [From Philippe DELEHAM, Dec 01 2008]
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. [From Jaroslav Krizek, Jul 27 2009]
a(n) = Sum_{k, 0<=k<=n} A201730(n,k)*3^k . - DELEHAM Philippe, Dec 06 2011
a(n) = Sum_{k, 0<=k<=n} A134309(n,k)*2^k = Sum_{k, 0<=k<=n} A055372(n,k). - DELEHAM Philippe, Feb 04 2012
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MAPLE
| seq(ceil(count(Subset(n))*count(Composition(n))/4), n=1..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 16 2006
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MATHEMATICA
| CoefficientList[Series[(1-2x)/(1-4x), {x, 0, 40}], x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* From Harvey P. Dale, Apr 22 2011 *)
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PROG
| (PARI) a(n)=1<<max(0, 2*n-1) \\ Charles R Greathouse IV, Jul 25 2011
(MAGMA) [(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011
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CROSSREFS
| Row sums of triangle A136158.
Cf. A081295, A009117. Essentially the same as A004171.
Cf. A136158, A016742.
Sequence in context: A067897 A145682 A099752 * A004171 A009117 A160637
Adjacent sequences: A081291 A081292 A081293 * A081295 A081296 A081297
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
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