A210068 Expansion of 1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)).

1, 2, 6, 12, 25, 44, 79, 128, 208, 318, 483, 704, 1019, 1430, 1992, 2712, 3664, 4862, 6407, 8320, 10735, 13686, 17344, 21760, 27153, 33592, 41353, 50532, 61468, 74290, 89415, 107008, 127576, 151332, 178882, 210496, 246898, 288420, 335920

Offset: 0

Author

F. Chapoton, Mar 17 2012

Keywords

• nonn
• easy

Comments

This is associated with the root system E7, and can be described using the additive function on the affine E7 diagram:

2

|

1--2--3--4--3--2--1

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2, 2, -4, -3, 0, 7, 4, -5, -4, -5, 4, 7, 0, -3, -4, 2, 2, -1).

Crossrefs

For G2, the corresponding sequence is A001399.

For F4, the corresponding sequence is A115264.

For E6, the corresponding sequence is A164680.

For E8, the corresponding sequence is A045513.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)) )); // G. C. Greubel, Jan 13 2020

Maple

seq(coeff(series(1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Jan 13 2020

Mathematica

CoefficientList[Series[1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{2,2,-4,-3,0,7,4,-5,-4,-5,4,7,0,-3,-4,2,2,-1},{1,2,6,12,25,44,79,128,208,318,483,704,1019,1430,1992,2712,3664,4862},40] (* Harvey P. Dale, Sep 24 2021 *)

PARI

Vec(1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4))+O(x^40)) \\ Charles R Greathouse IV, Sep 26 2012

Sage

x=PowerSeriesRing(QQ,'x',40).gen()
1/((1-x)^2*(1-x^2)^3*(1-x^3)^2*(1-x^4))

Formula

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