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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Alford Arnold, Aug 21 2009

Keywords

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

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

A164678 Convolve A008619 with sequences which map to 2,3,5,7,11,13,17... A000040 then, by bending when needed, summarize the results in a triangular array.

Original entry on oeis.org

1, 2, 1, 5, 4, 2, 9, 8, 6, 2, 17, 16, 14, 9, 3, 28, 27, 25, 20, 12, 3, 47, 46, 44, 39, 30, 16, 4, 73, 72, 70, 65, 56, 40, 20, 4, 114, 113, 111, 106, 97, 80, 55, 25, 5, 170, 169, 167, 162, 153, 136, 109, 70, 30, 5, 253, 252, 250, 245, 236, 219, 191, 147, 91, 36, 6, 365
Offset: 1

Views

Author

Alford Arnold, Aug 25 2009

Keywords

Comments

Sequence A164680 is not in the triangle since 45 = 3*15 and 15 is not a member of A000040 (the prime numbers).

Examples

			The desired triangle begins:
.1
.2..1
.5..4..2
.9..8..6..2
17.16.14..9..3
28.27.25.20.12..3
47.46.44.39.30.16..4
etc.
Note that 21 = 3*7 maps to 1,2,5,9,17,27,44,... A139672 is embedded in the triangle.

Crossrefs

Cf. A008619 A002620 A006918 A097701 A139672 ... A000097 (embedded sequences).
Showing 1-2 of 2 results.

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