A266755 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^4)).

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

Offset: 0

Author

N. J. A. Sloane, Jan 10 2016

Keywords

• nonn
• easy

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/Product_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. 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/Product_{i=2..k+1} (1-x^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

From Gus Wiseman, May 23 2021: (Start)

Also the number of integer partitions of n into at most n/2 parts, none greater than 3. The case of any maximum is A110618. The case of any length is A001399. The Heinz numbers of these partitions are given by A344293.

For example, the a(2) = 1 through a(13) = 5 partitions are:

2 3 22 32 33 322 332 333 3322 3332 3333 33322

31 222 331 2222 3222 3331 32222 33222 33331

321 3221 3321 22222 33221 33321 322222

3311 32221 33311 222222 332221

33211 322221 333211

332211

333111

(End)

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + ... - _Michael Somos_, Jan 29 2022
		

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, and 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).

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.

A variant of A005044.

Cf. A001400 (partial sums).

Cf. A308065.

Number of partitions of n whose Heinz number is in A344293.

A001399 counts partitions with all parts <= 3, ranked by A051037.

A025065 counts partitions of n with >= n/2 parts, ranked by A344296.

A035363 counts partitions of n with n/2 parts, ranked by A340387.

A110618 counts partitions of n into at most n/2 parts, ranked by A344291.

Cf. A000041, A008642, A279622, A325691, A344294, A344297.

Programs

Magma

I:=[1,0,1,1,2,1,3,2,4]; [n le 9 select I[n] else Self(n-2)+ Self(n-3)+Self(n-4)-Self(n-5)-Self(n-6)-Self(n-7)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Jan 11 2016

Mathematica

CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^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 *)
Table[Length[Select[IntegerPartitions[n],Length[#]<=n/2&&Max@@#<=3&]],{n,0,30}] (* Gus Wiseman, May 23 2021 *)
a[ n_] := Round[(n + 3*(2 - Mod[n,2]))^2/48]; (* Michael Somos, Jan 29 2022 *)

PARI

Vec(1/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jan 11 2016

PARI

{a(n) = round((n + 3*(2-n%2))^2/48)}; /* Michael Somos, Jan 29 2022 */

Sage

(1/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019

Formula

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. - Vincenzo Librandi, Jan 11 2016

a(n) = a(-9-n) for all n in Z. a(n) = a(n+3) for all n in 2Z. - Michael Somos, Jan 29 2022

E.g.f.: exp(-x)*(81 - 18*x + exp(2*x)*(107 + 60*x + 6*x^2) + 64*exp(x/2)*cos(sqrt(3)*x/2) + 36*exp(x)*(cos(x) - sin(x)))/288. - Stefano Spezia, Mar 05 2023

For n >= 3, if n is even, a(n) = a(n-3) + floor(n/4) + 1, otherwise a(n) = a(n-3). - Robert FERREOL, Feb 05 2024

a(n) = floor((n^2+9*n+(3*n+9)*(-1)^n+39)/48). - Hoang Xuan Thanh, Jun 03 2025