A164680 Expansion of x/((1-x)^3*(1-x^2)^3*(1-x^3)).

1, 3, 9, 20, 42, 78, 139, 231, 372, 573, 861, 1254, 1791, 2499, 3432, 4629, 6162, 8085, 10492, 13455, 17094, 21503, 26832, 33201, 40795, 49764, 60333, 72687, 87096, 103785, 123075, 145236, 170646, 199626, 232617, 269997, 312277, 359898, 413448, 473438

Offset: 1

Author

Alford Arnold, Aug 21 2009

Keywords

• nonn
• easy

Comments

Convolution of A006918 with A001399, or of A002625 with A059841 (A000035 if offsets are respected),

or of A038163 with A022003 or of A057524 with A027656 or of A014125 with the aerated version of A000217,

or of A002624 with A103221, or of A002623 with A008731, or of other combinations of splitting the signature -/3,3,1 into two components.

If we apply the enumeration of Molien series as described in A139672,

this is row 45=9*5 of a table of values related to Molien series, i.e., the

product of the sequence on row 9 (A006918) with the sequence on row 5 (A001399).

This is associated with the root system E6, and can be described using the additive function on the affine E6 diagram:

1

|

2

|

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

Examples

			To calculate a(3), we consider the first three terms of A001399 = (1 1 2...)
and the first three terms of A006918 = (1 2 5 ...), to get the convolved a(3) = 1*5+1*2+2*1 = 9.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (3,0,-7,3,6,0,-6,-3,7,0,-3,1).

Crossrefs

Cf. A139672 (row 21).

For G2, the corresponding sequence is A001399.

For F4, the corresponding sequence is A115264.

For E7, the corresponding sequence is A210068.

For E8, the corresponding sequence is A045513.

See A210634 for a closely related sequence.

Programs

Magma

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

Maple

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

Mathematica

Rest@CoefficientList[Series[x/((1-x)^3*(1-x^2)^3*(1-x^3)), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)
LinearRecurrence[{3,0,-7,3,6,0,-6,-3,7,0,-3,1},{1,3,9,20,42,78,139,231,372,573,861,1254},40] (* Harvey P. Dale, Aug 03 2025 *)

PARI

Vec(1/(1-x)^3/(1-x^2)^3/(1-x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

Sage

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

Formula

a(n) = round( -(-1)^n*(n+3)*(n+7)/256 +(6*n^6 +180*n^5 +2070*n^4 +11400*n^3 +30429*n^2 +34290*n +9785)/103680 ) - R. J. Mathar, Mar 19 2012

Extensions

Edited and extended by R. J. Mathar, Aug 22 2009

Corrected link to index entries - R. J. Mathar, Aug 26 2009