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A052950 Expansion of (2-3x-x^2+x^3)/((1-x)(1+x)(1-2x)). 4
2, 1, 3, 4, 9, 16, 33, 64, 129, 256, 513, 1024, 2049, 4096, 8193, 16384, 32769, 65536, 131073, 262144, 524289, 1048576, 2097153, 4194304, 8388609, 16777216, 33554433, 67108864, 134217729, 268435456, 536870913, 1073741824, 2147483649 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Equals row sums of triangle A178616 but replacing the 2 with a 1. - Gary W. Adamson, May 30 2010

Inverse binomial transform is (-1)^n * a(n). - Michael Somos, Jun 03 2014

LINKS

Table of n, a(n) for n=0..32.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1009

Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

FORMULA

G.f.: (x^3-x^2-3*x+2)/(-1+2*x)/(-1+x^2).

Recurrence: {a(1)=1, a(3)=4, a(2)=3, a(0)=2, -2*a(n)-a(n+1)+a(n+2)+1=0}

2^(n-1)+Sum(1/2*_alpha^(-n), _alpha=RootOf(-1+_Z^2))

E.g.f. : cosh(x)(1+exp(x)); a(n)=(2^n+1+(-1)^n+0^n)/2. - Paul Barry, Sep 18 2003

a(2*n + 1) = 4 * a(2*n - 1) for all n in Z. a(2*n + 2) = 3*a(2*n + 1) + 2*a(2*n) if n>0. - Michael Somos, Jun 04 2014

EXAMPLE

G.f. = 2 + x + 3*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 33*x^6 + 64*x^7 + 129*x^8 + ...

MAPLE

spec := [S, {S=Union(Sequence(Prod(Sequence(Z), Z)), Sequence(Prod(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

MATHEMATICA

f1[n_]:=2*n+1; f2[n_]:=2*(n-1); a=1; lst={a}; Do[AppendTo[lst, a=f1[a]]; AppendTo[lst, a=f2[a]], {n, 20}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)

a[ n_] := (2^n + 1 + (-1)^n + Boole[n == 0]) / 2; (* Michael Somos, Jun 03 2014 *)

a[ n_] := If[ n < 0, (1 - n)! SeriesCoefficient[ Sinh[x] + Exp[x/2], {x, 0, 1 - n}], n! SeriesCoefficient[ Cosh[x] (1 + Exp[x]), {x, 0, n}]]; (* Michael Somos, Jun 04 2014 *)

PROG

(PARI) {a(n) = (2^n + 1 + (-1)^n + (n==0)) / 2}; /* Michael Somos, Jun 03 2014 */

CROSSREFS

Cf. A178616. - Gary W. Adamson, May 30 2010

Sequence in context: A019612 A007444 A166476 * A086851 A001054 A218209

Adjacent sequences:  A052947 A052948 A052949 * A052951 A052952 A052953

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from James A. Sellers, Jun 05 2000

STATUS

approved

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Last modified April 24 12:21 EDT 2019. Contains 322429 sequences. (Running on oeis4.)