A008669 Molien series for 4-dimensional complex reflection group of order 7680 (in powers of x^4).
1, 1, 2, 3, 4, 6, 8, 10, 13, 16, 20, 24, 29, 34, 40, 47, 54, 62, 71, 80, 91, 102, 114, 127, 141, 156, 172, 189, 207, 226, 247, 268, 291, 315, 340, 367, 395, 424, 455, 487, 521, 556, 593, 631, 671, 713, 756, 801, 848, 896, 947, 999, 1053, 1109, 1167, 1227, 1289
Offset: 0
Examples
There are 6 partitions of 5 into parts 1,2,3 and 5. These are (5)(32)(311)(221)(2111)(11111). - _David Neil McGrath_, Sep 15 2014
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,3,5).
- L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 29).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 239
- Atsuto Seko, Atsushi Togo, and Isao Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, arXiv:1901.02118 [physics.comp-ph], 2019.
- Index entries for two-way infinite sequences
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,0,0,-1,0,1,1,-1).
- Index entries for Molien series
Programs
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Magma
[Round((n+3)*(2*n+9)*(n+9)/360): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011
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Maple
1/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)); seq(coeff(series(%, x, n+1), x, n), n = 0..60); # modified by G. C. Greubel, Sep 08 2019
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Mathematica
LinearRecurrence[{1,1,0,-1,0,0,-1,0,1,1,-1}, {1,1,2,3,4,6,8,10,13,16, 20}, 60] (* Harvey P. Dale, Feb 25 2015 *) -
PARI
a(n)=round((n+3)*(2*n+9)*(n+9)/360)
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Sage
[round((n+3)*(2*n+9)*(n+9)/360) for n in (0..60)] # G. C. Greubel, Sep 08 2019
Formula
a(n) = round((n+3)*(2*n+9)*(n+9)/360).
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)).
a(n) = -a(-11-n).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-7) + a(n-9) + a(n-10) - a(n-11). - David Neil McGrath, Sep 15 2014
Comments