A204200 INVERT transform of [1, 0, 1, 3, 9, 27, 81, ...].

1, 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: 1

Author

Michael Somos, Jan 12 2012

Keywords

• nonn

Comments

Same as A052544 except for beginning with an additional 1.

Number of permutations of length n>=0 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

			x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 60*x^6 + 189*x^7 + 595*x^8 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (4, -3, 1).

Crossrefs

Cf. A052544.

Programs

Haskell

a204200 n = a204200_list !! (n-1)
a204200_list = 1 : 1 : 2 : zipWith (+) a204200_list (tail $ zipWith (-)
   (map (* 4) (tail a204200_list)) (map (* 3) a204200_list))
-- Reinhard Zumkeller, Jan 16 2012

Mathematica

LinearRecurrence[{4, -3, 1}, {1, 1, 2}, 29] (* or *)
Rest@ CoefficientList[Series[-1 + 1/(1 - x - x^3/(1 - 3 x)), {x, 0, 29}], x] (* Michael De Vlieger, May 06 2019 *)

PARI

{a(n) = if( n<1, n = 1-n; polcoeff( (1 - x)^2 / (1 - 3*x + 4*x^2 - x^3) + x * O(x^n), n), polcoeff( x * (1 - 3*x + x^2) / (1 - 4*x + 3*x^2 - x^3) + x * O(x^n), n))}

Formula

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

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

a(n + 2) = A052544(n). That is, A052544 is the same except for extra 1 term and origin.

a(n+1) = Sum_{k, 0<=k<=n} A204533(n,k). - Philippe Deléham, Jan 18 2012