

A266755


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


21



1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120, 114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

This is the same as A005044 but without the three leading zeros. There are so many situations where one wants this sequence rather than A005044 that it seems appropriate for it to have its own entry.
But see A005044 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_3.
The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1x^(1+m_i)) where the m_i are [1,3,5,...,2k3,k1]. If k is even only even powers of x appear, and we bisect the sequence.
Also, Molien series for invariants of finite Coxeter group A_3. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1x^i). Note that this is the root system A_k not the alternating group Alt_k.
a(n) is the number of partitions of n into parts 2, 3, and 4.  Joerg Arndt, Apr 16 2017


REFERENCES

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sara C. Billey, Matjaž Konvalinka, Joshua P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,1,1,1,0,1).


FORMULA

a(n) = a(n2) + a(n3) + a(n4)  a(n5)  a(n6)  a(n7) + a(n9) for n>8.  Vincenzo Librandi, Jan 11 2016


MATHEMATICA

CoefficientList[Series[1/((1x^2)(1x^3)(1x^4)), {x, 0, 100}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{0, 1, 1, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 1, 2, 1, 3, 2, 4}, 100] (* Vincenzo Librandi, Jan 11 2016 *)


PROG

(PARI) Vec(1/((1x^2)*(1x^3)*(1x^4)) + O(x^100)) \\ Michel Marcus, Jan 11 2016
(MAGMA) I:=[1, 0, 1, 1, 2, 1, 3, 2, 4]; [n le 9 select I[n] else Self(n2)+ Self(n3)+Self(n4)Self(n5)Self(n6)Self(n7)+Self(n9): n in [1..100]]; // Vincenzo Librandi, Jan 11 2016
(Sage) (1/((1x^2)*(1x^3)*(1x^4))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019


CROSSREFS

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776A266781.
Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770A266775.
A variant of A005044.
Cf. A001400 (partial sums).
Cf. A308065.
Sequence in context: A325132 A225854 A005044 * A029142 A054685 A286220
Adjacent sequences: A266752 A266753 A266754 * A266756 A266757 A266758


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Jan 10 2016


STATUS

approved



