A266781 -id:A266781 - cp's OEIS frontend

A103221 Number of partitions of n into parts 2 and 3.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14, 15, 15

Offset: 0

Michael Somos, Jan 25 2005

Essentially the same as A008615.
Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - Andrew Baxter, Jun 28 2006
Also number of equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008
a(n) is also the number of 2-regular multigraphs on n vertices, where each component is either a pair of parallel edges, or a triangle. - Jason Kimberley, Oct 14 2011
For n>1, a(n) is the number of partitions of 2n into positive parts x,y, and z such that x>=y and y=z. This sequence is used in calculating the probability of the need for a run-off election when n voters randomly cast ballots for two of three candidates running for two empty slots on a county commission. - Dennis P. Walsh, Apr 25 2013
Also, Molien series for invariants of finite Coxeter group A_2. 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. - N. J. A. Sloane, Jan 11 2016
The coefficient of x^(2*n+1) in the power series expansion of the Weierstrass sigma function is a polynomial in the invariants g2 and g3 with a(n) terms. - Michael Somos, Jun 14 2016
a(n) is also the dimension of the complex vector space of modular forms M_{2*n} of weight 2*n and level 1 (full modular group). See Apostol p. 119, eq. (9) for k=2*n, and Ash and Gross, p. 178, Table 13.1. For a(6*k+1) = a(6*k+j)-1 for j = 0,2,3,4,5 and k >= 0 see A016921 (so-called dips, cf. Ash and Gross, p. 178.). - Wolfdieter Lang, Sep 16 2016
In an hexagonal tiling of the plane where the base tile is (0,0)--(2,1)--(3,3)--(1,4)--(-1,3)--(-2,1)--(0,0), a(n) is the number of vertices on the (n,0)--(n,n) closed line segment. - Luc Rousseau, Mar 22 2018

			For n=8, a(n)=2 since there are two partitions of 16 into 3 positive parts x, y, and z such that x >= y and y=z, namely, 16 = 8+4+4 and 16 = 6+5+5. - _Dennis P. Walsh_, Apr 25 2013
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - N. J. A. Sloane, Aug 28 2010.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 5.
Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
Ansgar Freyer, Monika Ludwig, and Martin Rubey, Unimodular Valuations beyond Ehrhart, arXiv:2407.07691 [math.MG], 2024. See p. 8.
Jun-Ichi Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Luc Rousseau, a(n) in an hexagonal tiling
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for Molien series
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776, A266777, A266778, A266779, A266780, A266781.
Cf. A004009, A013973, A016921, A079978.

[Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016

A103221:=n->floor((n+2)/2)-floor((n+2)/3): # Andrew Baxter, Jun 06 2008

a=b=c=d=0;Table[e=a+b-d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* Ray Chandler, Sep 23 2015 *)
a[ n_] := With[{m = Max[-5 - n, n]}, (-1)^Boole[n < 0] SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Jun 02 2019 *)

{a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};

a(n)=n+=2; n\2 - n\3 \\ Charles R Greathouse IV, Jul 31 2017

def A103221(n): return (n>>1)+1-(n+2)//3 # Chai Wah Wu, Apr 15 2025

def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = A008615(n+2). First differences of A001399.
a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3). - Andrew Baxter, Jun 06 2008
For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - Dennis P. Walsh, Apr 25 2013
a(n) = floor(n/6)+1 unless n == 1 (mod 6); if n == 1 (mod 6), a(n) = floor(n/6). - Bob Selcoe, Sep 27 2014
a(n) = A081753(2*n); see the Dennis P. Walsh formula. - Wolfdieter Lang, Sep 16 2016
a(n)-a(n-2) = A079978(n). - R. J. Mathar, Jun 23 2021
E.g.f.: (3*(4 + x)*cosh(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)) + 3*(1 + x)*sinh(x))/18. - Stefano Spezia, Mar 05 2023
a(n) = A008615(n-1)+A059841(n). - R. J. Mathar, May 03 2023

Name changed by Wolfdieter Lang, Sep 16 2016

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

Original entry on oeis.org

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

N. J. A. Sloane, Jan 10 2016

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)

			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

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

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).

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.

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

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 *)

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

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

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

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

A008667 Expansion of g.f.: 1/((1-x^2)(1-x^3)(1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 22, 23, 28, 29, 34, 36, 42, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 241, 258, 268, 286, 297, 316, 328, 348, 361, 382, 396, 419, 433, 457

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_4. 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. - N. J. A. Sloane, Jan 11 2016

Number of partitions into parts 2, 3, 4, and 5. - Joerg Arndt, Apr 29 2014

			a(4)=2 because f''''(x)/4!=2 at x=0 for f=1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)).
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 5*x^9 + 7*x^10 + 7*x^11 + ... .

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 32).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 241
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1,-2,-1,0,1,1,1,0,-1).

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

Cf. A005044, A001401 (partial sums).

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4)(1-x^5)),{x,0,#}]&/@Range[0,100] (* or *) a[k_]=SeriesCoefficient[1/((1-x^2)(1-x^3)(1-x^4) (1-x^5)),{x,0,k}] (* Peter Pein (petsie(AT)dordos.net), Sep 09 2006 *)
CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,5}],{x,0,70}],x] (* Harvey P. Dale, Feb 22 2018 *)

{a(n) = if( n<-13, -a(-14 - n), polcoeff( prod( k=2, 5, 1 / (1 - x^k), 1 + x * O(x^n)), n))} /* Michael Somos, Oct 14 2006 */

def A008667_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5))).list()
A008667_list(65) # G. C. Greubel, Sep 08 2019

Euler transform of length 5 sequence [ 0, 1, 1, 1, 1]. - Michael Somos, Sep 23 2006
a(-14 - n) = -a(n). - Michael Somos, Sep 23 2006
a(n) ~ 1/720*n^3. - Ralf Stephan, Apr 29 2014
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6) - 2*a(n-7) - a(n-8) + a(n-10) + a(n-11) + a(n-12) - a(n-14). - David Neil McGrath, Sep 13 2014
From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A008680(n).
a(n)-a(n-3) = A025802(n).
a(n)-a(n-4) = A025795(n).
a(n)-a(n-5) = A005044(n+3). (End)
a(n)= floor((n^3 + 21*n^2 + 156*n - 45*n*(n mod 2) + 720)/720 - [(n mod 10)=1]/5). - Hoang Xuan Thanh, Aug 20 2025

A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 16, 17, 23, 26, 33, 37, 47, 52, 64, 72, 86, 96, 115, 127, 149, 166, 192, 212, 245, 269, 307, 338, 382, 419, 472, 515, 576, 629, 699, 760, 843, 913, 1007, 1091, 1197, 1293, 1416, 1525, 1663, 1790, 1945, 2088, 2265, 2426

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_6. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

Cayley tabulates the coefficients in the expansion of H = 1 / ((1 - x^2) * (1 - x^4) * ... * (1 - x^14)) with even indices 0, 2, ..., 142.

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...
G.f. = 1 + q^2 + q^6 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^14 + 6*q^16 + ...

A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, American Journal of Mathematics, 2 (1879), pp.71-84. See pp.77-78.
A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -2, -2, -1, 0, 2, 2, 2, 2, 0, -1, -2, -2, -1, 0, 0, 1, 1, 1, 0, -1).

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

nn = 102; t = CoefficientList[Series[1/((1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^10)*(1 - x^12)*(1 - x^14)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

Euler transform of length 7 sequence [ 0, 1, 1, 1, 1, 1, 1]. - Michael Somos, Apr 23 2014

More terms from James Sellers, Feb 09 2000

A037145 Expansion of 1/((1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 9, 9, 14, 13, 19, 20, 26, 27, 36, 36, 47, 49, 60, 63, 78, 80, 97, 102, 120, 126, 149, 154, 180, 189, 216, 227, 260, 270, 307, 322, 361, 378, 424, 441, 492, 515, 568, 594, 656, 682, 750

Offset: 0

N. J. A. Sloane

Also, Molien series for invariants of finite Coxeter group A_5. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - N. J. A. Sloane, Jan 11 2016

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

Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 2, 2, 0, 0, -1, -1, -1, 0, 1).

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

Cf. A001402 (partial sums).

CoefficientList[Series[1/Times@@Table[(1-x^n),{n,2,6}],{x,0,50}],x] (* Harvey P. Dale, Dec 25 2012 *)

A266776 Molien series for invariants of finite Coxeter group A_7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 11, 12, 18, 19, 27, 30, 40, 44, 58, 64, 82, 91, 113, 126, 155, 171, 207, 230, 274, 303, 358, 395, 462, 509, 589, 649, 746, 818, 934, 1024, 1161, 1269, 1432, 1562, 1753, 1909, 2131, 2317, 2577, 2794, 3095, 3352, 3698, 3997, 4396, 4743, 5200, 5601, 6121, 6584, 7177, 7705, 8377, 8983, 9741, 10429, 11285, 12065

Offset: 0

N. J. A. Sloane, Jan 11 2016

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.

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -2, -2, -1, 1, 2, 2, 3, 2, 1, -1, -2, -3, -2, -2, -1, 1, 2, 2, 1, 1, 0, 0, -1, -1, -1, 0, 1).
Index entries for Molien series

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

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(&*[1-t^k: k in [2..8]]))); // G. C. Greubel, Oct 24 2018

CoefficientList[Series[1/Product[1-t^k, {k,2,8}], {t, 0, 40}], t] (* G. C. Greubel, Oct 24 2018 *)

t='t+O('t^40); Vec(1/prod(k=2,8, 1-t^k)) \\ G. C. Greubel, Oct 24 2018

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)).

A266777 Molien series for invariants of finite Coxeter group A_8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 51, 66, 75, 95, 110, 134, 155, 189, 215, 258, 296, 349, 398, 468, 529, 617, 698, 804, 907, 1042, 1167, 1332, 1492, 1690, 1886, 2130, 2366, 2660, 2951, 3298, 3649, 4069, 4484, 4981, 5482, 6064, 6657, 7347, 8041, 8849, 9670, 10605, 11565, 12659, 13769, 15034, 16330, 17782, 19278, 20955

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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.

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,0,-1,-1,-1,-2,-2,0,1,2,3,3,2,1,0,-2,-3,-4,-3,-2,0,1,2,3,3,2,1,0,-2,-2,-1,-1,-1,0,0,1,1,1,0,-1).
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (&*[1/(1-x^j): j in [2..9]]) )); // G. C. Greubel, Feb 01 2020

seq(coeff(series( mul(1/(1-x^j), j=2..9), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 01 2020

CoefficientList[Series[Product[1/(1-x^j), {j,2,9}], {x,0,70}], x] (* G. C. Greubel, Feb 01 2020 *)

Vec( prod(j=2,9, 1/(1-x^j)) + O('x^70) ) \\ G. C. Greubel, Feb 01 2020

def A266777_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..9)) ).list()
A266777_list(70) # G. C. Greubel, Feb 01 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)).

A266778 Molien series for invariants of finite Coxeter group A_9.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 48, 55, 73, 83, 107, 123, 154, 177, 220, 251, 306, 351, 422, 481, 575, 652, 771, 875, 1024, 1158, 1348, 1518, 1754, 1973, 2265, 2538, 2901, 3241, 3684, 4109, 4646, 5167, 5823, 6457, 7246, 8020, 8965, 9898, 11031, 12150, 13495, 14837, 16428, 18022, 19905, 21789, 23999, 26228, 28813

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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.

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -2, -1, 0, 1, 3, 3, 3, 2, 1, 0, -1, -4, -4, -4, -3, -2, 0, 2, 3, 4, 4, 4, 1, 0, -1, -2, -3, -3, -3, -1, 0, 1, 2, 1, 1, 1, 1, 0, 0, -1, -1, -1, 0, 1).
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( &*[1/(1-x^j): j in [2..10]] )); // G. C. Greubel, Feb 02 2020

seq(coeff(series( mul(1/(1-x^j), j=2..10), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 02 2020

CoefficientList[Series[Product[1/(1-x^j), {j,2,10}], {x,0,70}], x] (* G. C. Greubel, Feb 02 2020 *)
LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1,-1,-2,-1,0,1,3,3,3,2,1,0,-1,-4,-4,-4,-3,-2,0,2,3,4,4,4,1,0,-1,-2,-3,-3,-3,-1,0,1,2,1,1,1,1,0,0,-1,-1,-1,0,1},{1,0,1,1,2,2,4,4,7,8,12,13,20,22,31,36,48,55,73,83,107,123,154,177,220,251,306,351,422,481,575,652,771,875,1024,1158,1348,1518,1754,1973,2265,2538,2901,3241,3684,4109,4646,5167,5823,6457,7246,8020,8965,9898},70] (* Harvey P. Dale, Aug 10 2021 *)

Vec( prod(j=2,10, 1/(1-x^j)) +O('x^70) ) \\ G. C. Greubel, Feb 02 2020

def A266778_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( product(1/(1-x^j) for j in (2..10)) ).list()
A266778_list(70) # G. C. Greubel, Feb 02 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)*(1-t^10)).

A266779 Molien series for invariants of finite Coxeter group A_10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 23, 32, 38, 50, 59, 77, 90, 115, 135, 168, 197, 243, 283, 344, 401, 481, 558, 665, 767, 906, 1043, 1221, 1401, 1631, 1862, 2155, 2454, 2823, 3203, 3668, 4147, 4727, 5330, 6047, 6798, 7685, 8612, 9700, 10843, 12168, 13566, 15178, 16877, 18825, 20884, 23226, 25707, 28517, 31489, 34842, 38396

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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.

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 2, 3, 3, 3, 2, 1, 0, -2, -3, -4, -4, -5, -3, -1, 1, 3, 4, 5, 5, 4, 3, 1, -1, -3, -5, -4, -4, -3, -2, 0, 1, 2, 3, 3, 3, 2, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 1, 1, 1, 0, -1).
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^j: j in [2..11]]) )); // G. C. Greubel, Feb 03 2020

seq(coeff(series(1/mul(1-x^j, j=2..11), x, n+1), x, n), n = 0..70); # G. C. Greubel, Feb 03 2020

CoefficientList[Series[1/Product[1-x^j, {j,2,11}], {x,0,70}], x] (* G. C. Greubel, Feb 03 2020 *)

Vec( 1/prod(j=2,11,1-x^j) +O('x^70)) \\ G. C. Greubel, Feb 03 2020

def A266779_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product(1-x^j for j in (2..11))).list()
A266779_list(70) # G. C. Greubel, Feb 03 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)*(1-t^10)*(1-t^11)).

A266780 Molien series for invariants of finite Coxeter group A_11.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 23, 33, 39, 52, 61, 81, 94, 122, 143, 180, 211, 264, 306, 377, 440, 533, 619, 746, 861, 1028, 1186, 1401, 1612, 1895, 2168, 2532, 2894, 3356, 3822, 4414, 5008, 5755, 6516, 7448, 8410, 9580, 10780, 12232, 13737, 15524, 17388, 19592, 21885, 24580, 27400, 30674, 34117, 38097, 42269, 47074, 52133

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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.

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

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1, -1, -1, 0, -1, -1, 1, 2, 3, 3, 3, 2, 1, -1, -2, -3, -3, -5, -5, -4, -2, 0, 2, 4, 5, 6, 6, 5, 3, 2, -2, -3, -5, -6, -6, -5, -4, -2, 0, 2, 4, 5, 5, 3, 3, 2, 1, -1, -2, -3, -3, -3, -2, -1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, -1, -1, -1, 0, 1).
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^j: j in [2..12]]) )); // G. C. Greubel, Feb 04 2020

S:=series(1/mul(1-x^j, j=2..12)), x, 75):
seq(coeff(S, x, j), j=0..70); # G. C. Greubel, Feb 04 2020

CoefficientList[Series[1/Times@@(1-t^Range[2,12]),{t,0,70}],t] (* Harvey P. Dale, Jun 20 2017 *)

Vec( 1/prod(j=2,12, 1-x^j) +O('x^70) ) \\ G. C. Greubel, Feb 04 2020

def A266780_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^j for j in (2..12)) ).list()
A266780_list(70) # G. C. Greubel, Feb 04 2020

G.f.: 1/((1-t^2)*(1-t^3)*(1-t^4)*(1-t^5)*(1-t^6)*(1-t^7)*(1-t^8)*(1-t^9)*(1-t^10)*(1-t^11)*(1-t^12)).

cp's OEIS Frontend

A103221 Number of partitions of n into parts 2 and 3.

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

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A008667 Expansion of g.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)).

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A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

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A037145 Expansion of 1/((1-x^2)(1-x^3)...(1-x^6)).

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A266776 Molien series for invariants of finite Coxeter group A_7.

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

A008667 Expansion of g.f.: 1/((1-x^2)(1-x^3)(1-x^4)*(1-x^5)).