A052544 Expansion of (1-x)^2/(1 - 4*x + 3*x^2 - x^3).

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

Offset: 0

Author

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

Keywords

• easy
• nonn

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)

a(n-1) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements, and the fourth element is larger than the second element. - Sergey Kitaev, Dec 09 2020

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
• Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 480
• Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
• H. Stephan, Rekursive Folgen im Pascalschen Dreieck
• Index entries for linear recurrences with constant coefficients, signature (4,-3,1).

Crossrefs

Cf. A124820 (partial sums).

Cf. A052529, A001519.

Programs

GAP

a:=[1,2,6];; for n in [4..30] do a[n]:=4*a[n-1]-3*a[n-2]+a[n-3]; od; a; # G. C. Greubel, May 09 2019

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..30]]; // Vincenzo Librandi, Jan 12 2012

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, 30}] (* or *)
CoefficientList[Series[(1-x)^2/(1-4x+3x^2-x^3), {x, 0, 30}], x] (* Michael De Vlieger, Aug 03 2016 *)

PARI

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

PARI

Vec((1-x)^2/(1-4*x+3*x^2-x^3)+O(x^30)) \\ Charles R Greathouse IV, Jan 12 2012

Sage

((1-x)^2/(1-4*x+3*x^2-x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019

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).

a(n) = 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 L. Ollerton, 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

Extensions

More terms from James Sellers, Jun 06 2000