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A052544 Expansion of (1-x)^2/(1-4x+3x^2-x^3). 6
1, 2, 6, 19, 60, 189, 595, 1873, 5896, 18560, 58425, 183916, 578949, 1822473, 5736961, 18059374, 56849086, 178955183, 563332848, 1773314929, 5582216355, 17572253481, 55315679788, 174128175064, 548137914373, 1725482812088 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals INVERT transform of (1, 1, 3, 8, 21, 55, 144,...). [Gary W. Adamson, May 01 2009]

The Ze2 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence. [Johannes W. Meijer, Aug 16 2011]

Equals the partial sums of A052529 starting (1, 1, 4, 13, 41, 129,...). - Gary W. Adamson, Feb 15 2012

First trisection of Narayana's cows sequence A000930. - Oboifeng Dira, Aug 03 2016

From Peter Bala, Nov 03 2017: (Start)

Let f(x) = x/(1 - x^3), the characteristic function of numbers of the form 3*n + 1. Then f(f(x)) = Sum_{n >= 0} a(n)*x^(3*n+1).

a(n) = the number of compositions of 3*n + 1 into parts of the form 3*m + 1. For example, a(2) = 6 and the six compositions of 7 into parts of the form 3*m + 1 are 7, 4 + 1 + 1 + 1, 1 + 4 + 1 + 1, 1 + 1 + 4 + 1, 1 + 1 + 1 + 4 and 1 + 1 + 1 + 1 + 1 + 1 + 1. Cf. A001519, which gives the number of compositions of an odd number into odd parts. (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..500

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 480

H. Stephan, Rekursive Folgen im Pascalschen Dreieck

Index entries for linear recurrences with constant coefficients, signature (4,-3,1).

FORMULA

G.f.: -(-1+x)^2/(-1+4*x-3*x^2+x^3).

a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3).

Sum(-1/31*(-4-7*_alpha+2*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+4*_Z-3*_Z^2+_Z^3)).

a(n) = Sum_{k=0..n} binomial(n+2*k, 3*k), - Richard Ollerton (r.ollerton(AT)uws.edu.au), May 12 2004

G.f.: 1 / (1 - x - x / (1 - x)^2). - Michael Somos, Jan 12 2012

a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -4/27). - Peter Luschny, Nov 03 2017

EXAMPLE

G.f. = 1 + 2*x + 6*x^2 + 19*x^3 + 60*x^4 + 189*x^5 + 595*x^6 + 1873*x^7 + ...

MAPLE

spec := [S, {S=Sequence(Union(Z, Prod(Z, Sequence(Z), Sequence(Z))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..25);

A052544 := proc(n): add(binomial(n+2*k, 3*k), k=0...n)  end: seq(A052544(n), n=0..25); # Johannes W. Meijer, Aug 16 2011

MATHEMATICA

LinearRecurrence[{4, -3, 1}, {1, 2, 6}, 30] (* Harvey P. Dale, Jul 13 2011 *)

Table[Sum[Binomial[n + 2 k, 3 k], {k, 0, n}], {n, 0, 25}] (* or *)

CoefficientList[Series[-(-1 + x)^2/(-1 + 4 x - 3 x^2 + x^3), {x, 0, 25}], x] (* Michael De Vlieger, Aug 03 2016 *)

PROG

(PARI) {a(n) = sum(k=0, n, binomial(n + 2*k, 3*k))}; /* Michael Somos, Jan 12 2012 */

(MAGMA) I:=[1, 2, 6]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 12 2012

(PARI) Vec((1-x)^2/(1-4*x+3*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

CROSSREFS

Cf. A124820 (partial sums).

Cf. A052529, A001519.

Sequence in context: A294500 A208481 * A204200 A318127 A001169 A187276

Adjacent sequences:  A052541 A052542 A052543 * A052545 A052546 A052547

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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Last modified October 16 00:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)