A212393 Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.

1, 1, 2, 5, 14, 30, 72, 195, 485, 1059, 2065, 3682, 6120, 9620, 14454, 20925, 29367, 40145, 53655, 70324, 90610, 115002, 144020, 178215, 218169, 264495, 317837, 378870, 448300, 526864, 615330, 714497, 825195, 948285, 1084659, 1235240, 1400982, 1582870, 1781920

Offset: 0

Author

Bruno Berselli, May 14 2012

Keywords

• nonn
• easy

Comments

In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(0,k,0,0)(x,0) represents a generating function depending on k and x.

For successive values of k we have:

k=1, the g.f. of A000012: 1/(1-x);

k=2, the g.f. of A028310: (1-x+x^2)/(1-x)^2;

k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);

k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).

This sequence corresponds to the case k=5.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (page 21, Theorem 10).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

m:=39; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));

Mathematica

CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]

PARI

Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))

Formula

G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.

a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.

G.f.: 1+x+2*x^2+5*x^3 + 14*x^4*G(0), where G(k)= 1 + x*(k+1)*(124*k^3+192*k^2+89*k+180)/( (2*k+1)*(62*k^3+3*k^2-5*k+84) - x*(2*k+1)*(62*k^3+3*k^2-5*k+84)*(2*k+3)*(62*k^3+189*k^2+187*k+144)/(x*(2*k+3)*(62*k^3+189*k^2+187*k+144) + (k+1)*(124*k^3+192*k^2+89*k+180)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013