A125187 Number of Dumont permutations of the first kind of length 2n avoiding the patterns 1423 and 4132.
1, 1, 3, 12, 52, 232, 1049, 4777, 21845, 100159, 460023, 2115350, 9735205, 44829766, 206526972, 951759621, 4387156587, 20226421380, 93264500832, 430091815527, 1983549213861, 9148582037193, 42197572190160, 194643215702835
Offset: 0
Keywords
Examples
G.f. = 1 + x + 3*x^2 + 12*x^3 + 52*x^4 + 232*x^5 + 1049*x^6 + 4777*x^7 + 21845*x^8 + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Cyril Banderier, Michael Wallner, Lattice paths with catastrophes, arXiv:1707.01931 [math.CO], 2017.
- Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
- A. Burstein, Restricted Dumont permutations, arXiv:math/0402378 [math.CO], 2004
- A. Burstein, Restricted Dumont permutations, Annals of Combinatorics, 9, 2005, 269-280 (Theorem 3.12).
Programs
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Maple
C:=(1-sqrt(1-4*x))/2/x: G:=(2-(1+x)*C)/(2-x-(1+x)*C): Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=0..26);
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Mathematica
a[ n_] := SeriesCoefficient[ (2 - 9 x + x^2 + (x + x^2) Sqrt[1 - 4 x]) / (2 (1 - 5 x + 2 x^2 - x^3)), {x, 0, n}]; (* Michael Somos, Jan 14 2014 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( (2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x + x * O(x^n)) ) / (2 * (1 - 5*x + 2*x^2 - x^3)), n))}; /* Michael Somos, Jan 14 2014 */
Formula
G.f.: [2-(1+x)C(x)]/[2-x-(1+x)C(x)], where C(x)=(1-sqrt(1-4x))/(2x) is the Catalan function.
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) = upper left term in M^n, where M is an infinite square production matrix in which two columns of (1,2,3,...) are prepended to an infinite lower triangular matrix of all 1's and the rest zeros, as follows:
1, 1, 0, 0, 0, 0, ...
2, 2, 1, 0, 0, 0, ...
3, 3, 1, 1, 0, 0, ...
4, 4, 1, 1, 1, 0, ...
5, 5, 1, 1, 1, 1, ...
... (End)
Given g.f. A(x), then 0 = A(x)^2 * (x^3 - 2*x^2 + 5*x - 1) + A(x) *(x^2 - 9*x + 2) + (x^2 + 4*x -1). - Michael Somos, Jan 14 2014
0 = a(n)*(16*a(n+1) +6*a(n+2) -14*a(n+3) +210*a(n+4) -128*a(n+5) +18*a(n+6)) +a(n+1)*(-46*a(n+1) +143*a(n+2) -173*a(n+3) -283*a(n+4) +202*a(n+5) -29*a(n+6)) +a(n+2)*(-63*a(n+2) +386*a(n+3) +765*a(n+4) -529*a(n+5) +75*a(n+6)) +a(n+3)*(-559*a(n+3) +509*a(n+4) -149*a(n+5) +19*a(n+6)) +a(n+4)*(-108*a(n+4) +71*a(n+5) -12*a(n+6)) +a(n+5)*(-4*a(n+5) +a(n+6)). - Michael Somos, Jan 14 2014
G.f.: ( 2 - 9*x + x^2 + (x + x^2) * sqrt(1 - 4*x) ) / (2 - 10*x + 4*x^2 - 2*x^3). - Michael Somos, Apr 15 2012
G.f. = (1 - 3*y + y^2) / (1 - 4*y + 3*y^2 - y^3) = 1 / (1 - y / (1 - y / (1 - 2*y / (1 + y / (2 - y))))) where y = (1 - sqrt(1 - 4*x)) / 2. - Michael Somos, Apr 12 2012
D-finite with recurrence (-n+1)*a(n) +4*(2*n-3)*a(n-1) +(-13*n+19)*a(n-2) +(-13*n+75)*a(n-3) +(5*n-29)*a(n-4) +2*(-2*n+9)*a(n-5)=0. - R. J. Mathar, Jul 27 2013
Comments