A266769 -id:A266769 - cp's OEIS frontend

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

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

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

A003402 G.f.: 1/((1-x)(1-x^2)(1-x^3)^2(1-x^4)(1-x^5)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 19, 27, 37, 49, 64, 84, 106, 134, 168, 207, 253, 309, 371, 445, 530, 626, 736, 863, 1003, 1163, 1343, 1543, 1766, 2017, 2291, 2597, 2935, 3305, 3712, 4161, 4647, 5181, 5763, 6394, 7079, 7825, 8627, 9497, 10436, 11445, 12531, 13702, 14952

Offset: 0

N. J. A. Sloane

Enumerates certain triangular arrays of integers.

Also, Molien series for invariants of finite Coxeter group D_6 (bisected). The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_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. - N. J. A. Sloane, Jan 11 2016

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 0..10000
J. C. P. Miller, On the enumeration of partially ordered sets of integers, pp. 109-124 of T. P. McDonough and V. C. Mavron, editors, Combinatorics: Proceedings of the Fourth British Combinatorial Conference 1973. London Mathematical Society, Lecture Note Series, Number 13, Cambridge University Press, NY, 1974. The g.f. is in Eq. (8.1).
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, -1, -2, -1, -1, 2, 2, 2, -1, -1, -2, -1, 1, 1, 1, -1).

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

Cf. A003403, A003404, A003405, A029073, A256975, A256976, A256977.

A079978:= n -> `if`(n mod 3 = 0, 1, 0):
F:= n -> 1+floor((7913/17280)*n+(13/96)*n^2+(227/12960)*n^3+(1/960)*n^4+(1/43200)*n^5 + n/27*A079978(n) + n/128*(-1)^n):
seq(F(n), n= 0..100); # Robert Israel, Apr 22 2015

CoefficientList[Series[1/((1 - x) (1 - x^2) (1 - x^3)^2*(1 - x^4) (1 - x^5)), {x, 0, 49}], x] (* Michael De Vlieger, Feb 21 2018 *)

Vec(1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020

a(n) = a(n-1) + b(n), b(n) = b(n-2) + c(n) - e(n), c(n) = c(n-3) + 2e(n), e(n) = e(n - 4) + f(n), f(n) = f(n - 5) + g(n), g(n) = g(n - 6), g(0) = 1, all functions are 0 for negative indexes. [From Miller paper.] - Sean A. Irvine, Apr 22 2015

a(n) = 1 + floor((7913/17280)*n + (13/96)*n^2 + (227/12960)*n^3 + (1/960)*n^4 + (1/43200)*n^5 + n/27*A079978(n) + n/128*(-1)^n). - Robert Israel, Apr 22 2015

Entry revised by N. J. A. Sloane, Apr 22 2015

A266768 Molien series for invariants of finite Coxeter group D_5.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 3, 1, 5, 2, 7, 3, 10, 5, 13, 7, 18, 10, 23, 13, 30, 18, 37, 23, 47, 30, 57, 37, 70, 47, 84, 57, 101, 70, 119, 84, 141, 101, 164, 119, 192, 141, 221, 164, 255, 192, 291, 221, 333, 255, 377, 291, 427, 333, 480, 377, 540, 427, 603, 480, 674, 540, 748, 603, 831, 674, 918, 748, 1014, 831, 1115, 918, 1226, 1014, 1342, 1115

Offset: 0

N. J. A. Sloane, Jan 10 2016

nonn

The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_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.

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

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

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) )); // G. C. Greubel, Jan 31 2020

seq(coeff(series(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Jan 31 2020

CoefficientList[Series[1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)), {x,0,80}], x] (* G. C. Greubel, Jan 31 2020 *)

my(x='x+O('x^80)); Vec(1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8))) \\ G. C. Greubel, Jan 31 2020

def A266768_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^5)*(1-x^8)) ).list()
A266768_list(80) # G. C. Greubel, Jan 31 2020

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

a(n) = a(n-2)+a(n-4)+a(n-5)-a(n-7)-a(n-9)-2*a(n-10)+2*a(n-15)+a(n-16)+a(n-18)-a(n-20)-a(n-21)-a(n-23)+a(n-25). - Wesley Ivan Hurt, May 03 2021

A266770 Molien series for invariants of finite Coxeter group D_7.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 5, 1, 7, 2, 11, 3, 15, 5, 21, 7, 28, 11, 38, 15, 49, 21, 65, 28, 82, 38, 105, 49, 131, 65, 164, 82, 201, 105, 248, 131, 300, 164, 364, 201, 436, 248, 522, 300, 618, 364, 733, 436, 860, 522, 1009, 618, 1175, 733, 1367, 860, 1579, 1009, 1824, 1175, 2093, 1367, 2400, 1579, 2738, 1824, 3120, 2093, 3539, 2400, 4011

Offset: 0

N. J. A. Sloane, Jan 10 2016

nonn

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

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

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^7)*(&*[1-x^(2*j): j in [1..6]])) )); // G. C. Greubel, Jan 31 2020

seq(coeff(series(1/((1-x^7)*mul(1-x^(2*j), j=1..6)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Jan 31 2020

CoefficientList[Series[1/((1-x^7)*Product[1-x^(2*j), {j,6}]), {x,0,80}], x] (* G. C. Greubel, Jan 31 2020 *)

Vec(1/((1-x^7)*prod(j=1,6,1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Jan 31 2020

def A266770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^7)*product(1-x^(2*j) for j in (1..6))) ).list()
A266770_list(80) # G. C. Greubel, Jan 31 2020

G.f.: 1/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^7)*(1-x^8)*(1-x^10)*(1-x^12)).

A266775 Molien series for invariants of finite Coxeter group D_12 (bisected).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 24, 33, 47, 63, 88, 115, 155, 202, 266, 341, 443, 560, 715, 897, 1129, 1401, 1746, 2146, 2645, 3228, 3941, 4771, 5781, 6948, 8353, 9979, 11913, 14144, 16785, 19814, 23374, 27454, 32211, 37645, 43954, 51130, 59417, 68827, 79631, 91863, 105857, 121645

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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.

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
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^6)*(&*[1-x^j: j in [1..11]])) )); // G. C. Greubel, Jan 31 2020

S:=series(1/((1-x^6)*mul(1-x^j, j=1..11)), x, 55): seq(coeff(S, x, j), j=0..50); # G. C. Greubel, Jan 31 2020

CoefficientList[Series[1/((1-t^6)*Product[1-t^j, {j,11}]), {t,0,50}], t] (* G. C. Greubel, Jan 31 2020 *)

Vec( 1/( (1-x^6)*prod(j=1,11, 1-x^j) ) + O('x^50)) \\ G. C. Greubel, Jan 31 2020

[( 1/((1-x^6)*product(1-x^j for j in (1..11))) ).series(x, n+1).list()[n] for n in (0..50)] # G. C. Greubel, Jan 31 2020

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)*(1-t^12)^2*(1-t^14)*(1-t^16)*(1-t^18)*(1-t^20)*(1-t^22)), bisected.

G.f.: 1/( (1-t^6)*Product_{j=1..11} (1-t^j) ). - G. C. Greubel, Feb 01 2020

A060999 Nearest integer to (n+1)^3/9.

Original entry on oeis.org

0, 1, 3, 7, 14, 24, 38, 57, 81, 111, 148, 192, 244, 305, 375, 455, 546, 648, 762, 889, 1029, 1183, 1352, 1536, 1736, 1953, 2187, 2439, 2710, 3000, 3310, 3641, 3993, 4367, 4764, 5184, 5628, 6097, 6591, 7111, 7658, 8232, 8834, 9465, 10125, 10815, 11536

Offset: 0

N. J. A. Sloane, May 14 2001

a(n) is also the number of ways to award 4n+5 bonuses to 4 teams: first, second, third and fourth satisfying 1st > 2nd > 3rd > 4th and 1st + 4th < 2nd + 3rd. - Hoang Xuan Thanh, Jun 03 2025

			x + 3*x^2 + 7*x^3 + 14*x^4 + 24*x^5 + 38*x^6 + 57*x^7 + 81*x^8 + ...

Harry J. Smith, Table of n, a(n) for n=0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A266769, A008763.

Table[Floor[(n+1)^3/9+1/2],{n,0,50}] (* Harvey P. Dale, Jan 20 2013 *) (* or *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 1, 3, 7, 14, 24}, 47] (* Georg Fischer, Oct 13 2020 *)

a(n) = { round((n + 1)^3/9) } \\ Harry J. Smith, Jul 16 2009

{a(n) = n++; (n^3 - kronecker(-3, n)) / 9} /* Michael Somos, Aug 12 2009 */

G.f.: x*(1+x^2)/((1-x)^3*(1-x^3)).
G.f.: x * (1 - x^4) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)).
G.f.: ( (1 + 4*x + x^2) / (1 - x)^4 - 1 / (1 + x + x^2) ) / 9.
From Michael Somos, Aug 12 2009: (Start)
Euler transform of length 4 sequence [ 3, 1, 1, -1].
a(-2-n) = -a(n). (End)
E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(1 + 7*x + 6*x^2 + x^3) - 3*cos(sqrt(3)*x/2)+ sqrt(3)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Sep 24 2024
From Hoang Xuan Thanh, Jun 03 2025: (Start)
a(n) = floor(((n+1)^3+1)/9).
For n>0: a(n) = A266769(2n-2). (End)

A266772 Molien series for invariants of finite Coxeter group D_9.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 1, 7, 1, 11, 2, 15, 3, 22, 5, 30, 7, 41, 11, 54, 15, 73, 22, 94, 30, 123, 41, 157, 54, 201, 73, 252, 94, 318, 123, 393, 157, 488, 201, 598, 252, 732, 318, 887, 393, 1076, 488, 1291, 598, 1549, 732, 1845, 887, 2194, 1076, 2592, 1291, 3060, 1549, 3589, 1845, 4206, 2194, 4904, 2592, 5708, 3060, 6615, 3589

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^9)*(&*[1-x^(2*j): j in [1..8]])) )); // G. C. Greubel, Feb 03 2020

seq(coeff(series(1/((1-x^9)*mul(1-x^(2*j), j=1..8)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Feb 03 2020

CoefficientList[Series[1/((1-x^9)*Product[1-x^(2*j), {j,8}]), {x,0,80}], x] (* G. C. Greubel, Feb 03 2020 *)

Vec(1/((1-x^9)*prod(j=1,8,1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Feb 03 2020

def A266772_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^9)*product(1-x^(2*j) for j in (1..8))) ).list()
A266772_list(80) # G. C. Greubel, Feb 03 2020

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^9)*(1-t^10)*(1-t^12)*(1-t^14)*(1-t^16)).

A266773 Molien series for invariants of finite Coxeter group D_10 (bisected).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 49, 66, 90, 119, 158, 206, 267, 342, 437, 551, 694, 865, 1074, 1324, 1627, 1985, 2414, 2919, 3518, 4219, 5045, 6003, 7125, 8422, 9927, 11660, 13660, 15949, 18578, 21575, 24998, 28884, 33303, 38298, 43955, 50329, 57513, 65581, 74645, 84786

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1-x^5)*(&*[1-x^j: j in [1..9]])) )); // G. C. Greubel, Feb 03 2020

seq(coeff(series(1/((1-x^5)*mul(1-x^j, j=1..9)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Feb 03 2020

CoefficientList[Series[1/((1-x^5)*Product[1-x^j, {j,9}]), {x,0,50}], x] (* G. C. Greubel, Feb 03 2020 *)

Vec(1/((1-x^5)*prod(j=1,9,1-x^j)) +O('x^50)) \\ G. C. Greubel, Feb 03 2020

def A266773_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^5)*product(1-x^j for j in (1..9))) ).list()
A266773_list(50) # G. C. Greubel, Feb 03 2020

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)^2*(1-t^12)*(1-t^14)*(1-t^16)*(1-t^18)), bisected.

G.f.: 1/( (1-x^5)*(Product_{j=1..9} 1-x^j) ). - G. C. Greubel, Feb 03 2020

A266774 Molien series for invariants of finite Coxeter group D_11.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 1, 11, 1, 15, 2, 22, 3, 30, 5, 42, 7, 56, 11, 76, 15, 99, 22, 131, 30, 169, 42, 219, 56, 278, 76, 355, 99, 445, 131, 560, 169, 695, 219, 863, 278, 1060, 355, 1303, 445, 1586, 560, 1930, 695, 2331, 863, 2812, 1060, 3370, 1303, 4035, 1586, 4802, 1930, 5708, 2331, 6751, 2812, 7972, 3370, 9373, 4035, 11004

Offset: 0

N. J. A. Sloane, Jan 11 2016

nonn

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
Index entries for Molien series

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

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^11)*(&*[1-x^(2*j): j in [1..10]])) )); // G. C. Greubel, Feb 03 2020

seq(coeff(series(1/((1-x^11)*mul(1-x^(2*j), j=1..10)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Feb 03 2020

CoefficientList[Series[1/((1-x^11)*Product[1-x^(2*j), {j,10}]), {x,0,80}], x] (* G. C. Greubel, Feb 03 2020 *)

Vec(1/((1-x^11)*prod(j=1,10,1-x^(2*j))) +O('x^80)) \\ G. C. Greubel, Feb 03 2020

def A266774_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x^11)*product(1-x^(2*j) for j in (1..10))) ).list()
A266774_list(80) # G. C. Greubel, Feb 03 2020

G.f.: 1/((1-t^2)*(1-t^4)*(1-t^6)*(1-t^8)*(1-t^10)*(1-t^11)*(1-t^12)*(1-t^14)*(1-t^16)*(1-t^18)*(1-t^20)).

cp's OEIS Frontend

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

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A008763 Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

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A003402 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)).

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A266768 Molien series for invariants of finite Coxeter group D_5.

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A266770 Molien series for invariants of finite Coxeter group D_7.

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

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

A003402 G.f.: 1/((1-x)(1-x^2)(1-x^3)^2(1-x^4)(1-x^5)).