A103221 -id:A103221 - cp's OEIS frontend

A140953 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)*(1-x^13)).

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 25, 28, 32, 36, 41, 46, 52, 58, 65, 72, 80, 89, 98, 109, 119, 132, 144, 158, 173, 189, 206, 224, 244, 264, 287, 310, 336, 362, 391, 421, 453, 487, 523, 561, 601, 644, 688, 736, 785, 838, 893

Offset: 0

Alois P. Heinz, Jul 25 2008

Number of partitions of n into the first 6 primes. [Corrected by Harvey P. Dale, Dec 05 2022]

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0, 0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1).

Cf. A000040, A025795, A029144, A103221, A140952, A335106.

M := Matrix(41, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 3, 11, 19, 20, 21, 22, 30, 38, 39])) then 1 elif j=1 and member(i, [8, 9, 16, 17, 24, 25, 32, 33, 41]) then -1 else 0 fi):
a:= n -> (M^(n))[1,1]:
seq(a(n), n=0..50);

CoefficientList[Series[1/Times@@Table[1-x^p,{p,Prime[Range[6]]}],{x,0,60}],x] (* or *) LinearRecurrence[{0,1,1,0,0,0,0,-1,-1,0,1,0,0,0,0,-1,-1,0,1,1,1,1,0,-1,-1,0,0,0,0,1,0,-1,-1,0,0,0,0,1,1,0,-1},{1,0,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,16,19,21,25,28,32,36,41,46,52,58,65,72,80,89,98,109,119,132,144,158,173,189,206},70] (* Harvey P. Dale, Dec 05 2022 *)

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

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)).

A376782 Triangle read by rows: T(n,m) is the number of unlabeled graphs with n vertices having m minimum forbidden subgraphs, n >= 1, 1 <= m <= A371162(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 4, 2, 1, 4, 8, 13, 8, 1, 4, 5, 7, 20, 34, 31, 28, 12, 8, 5, 0, 1, 1, 4, 5, 13, 26, 33, 43, 59, 50, 62, 58, 60, 64, 67, 63, 70, 68, 65, 61, 60, 31, 28, 16, 8, 13, 4, 4, 4, 0, 2, 1, 0, 1, 1, 4, 6, 9, 21, 34, 39, 71, 74, 77, 99, 118, 124, 107, 129

Offset: 1

Max Alekseyev, Oct 03 2024

			Triangle starts with
n = 1: 1
n = 2: 1 1
n = 3: 1 3
n = 4: 1 4 4  2
n = 5: 1 4 8 13  8
n = 6: 1 4 5  7 20 34 31 28 12 8 5 0 1
...

Max Alekseyev, Table of k, a(k) for k = 1..209 (rows n = 1..8 flattened)
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, 2024.

Cf. A000088 (row sums), A371162 (row lengths), A000012 (column m=1), A113311 (column m=2).

Cf. A002865, A006897, A103221, A371161, A376780, A376781.

A081753 a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9

Offset: 0

Benoit Cloitre, Apr 08 2003

a(2n) = dimension of M(2n), where M(2n) denotes the complex vector space of modular forms of weight 2n for the group : PSL2(Z). dimension of M(2n+1) = 0.
See A103221(n) for the dimension of M(2n). The Apostol reference, p. 119, eq. (9) uses even k. - Wolfdieter Lang, Sep 16 2016
The space of modular forms is generated by E_4 (A004009) and E_6 (A013973) both of even weight. This is why the space of modular forms of odd weight is trivial. - Michael Somos, Dec 11 2018

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + ... - _Michael Somos_, Dec 11 2018

Apostol, Tom M., Modular Functions and Dirichlet Series in Number Theory, second edition, Springer, 1990.
Yves Hellegouarch, "Invitation aux mathématiques de Fermat-Wiles", Dunod, 2ème édition, p. 285

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A008615, A097992, A103221.

seq(floor(n/12)+1-charfcn[0](n-2 mod 12), n=0..100); # Robert Israel, Sep 16 2016

Table[If[Mod[n, 12] == 2, Floor[n/12], Floor[n/12] + 1], {n, 0, 120}] (* or *)
CoefficientList[Series[(1 - x^2 + x^3)/(1 - x - x^12 + x^13), {x, 0, 120}], x] (* Michael De Vlieger, Sep 19 2016 *)
a[ n_] := Quotient[n, 12] + Boole[Mod[n, 12] != 2]; (* Michael Somos, Dec 11 2018 *)

a(k) = if(k%12-2, floor(k/12)+1, floor(k/12))

{a(n) = n\12 + (n%12!=2)}; /* Michael Somos, Dec 11 2018 */

a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.
G.f.: (1-x^2+x^3)/(1-x-x^12+x^13). - Robert Israel, Sep 16 2016
a(2*n) = A008615(n+2), a(2*n+1) = A097992(n). - Michael Somos, Dec 11 2018

A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 4, 2, 4, 6, 4, 6, 9, 6, 9, 12, 9, 12, 16, 12, 16, 20, 16, 20, 25, 20, 25, 30, 25, 30, 36, 30, 36, 42, 36, 42, 49, 42, 49, 56, 49, 56, 64, 56, 64, 72, 64, 72, 81, 72, 81, 90, 81, 90, 100, 90, 100, 110, 100, 110, 121, 110, 121, 132

Offset: 0

Andrew Ivashenko, Mar 19 2019

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

Cf. A000041, A002620, A103221, A114209, A321202.

Cf. A008133, A069905, A008611.

a:=[0,0,0,1,0,1,2,1];; for n in [9..80] do a[n]:=a[n-2]+2*a[n-3] -2*a[n-5]-a[n-6]+a[n-8]; od; a; # G. C. Greubel, Apr 03 2019

R:=PowerSeriesRing(Integers(), 80); [0,0,0] cat Coefficients(R!( x^3/((1-x^2)*(1-x^3)^2) )); // G. C. Greubel, Apr 03 2019

LinearRecurrence[{0,1,2,0,-2,-1,0,1}, {0,0,0,1,0,1,2,1}, 80] (* G. C. Greubel, Apr 03 2019 *)
Table[(6n(2+n)-5-27(-1)^n+8(4+3n)Cos[2n Pi/3]-8Sqrt[3]n Sin[2n Pi/3])/216,{n,0,66}] (* Stefano Spezia, Apr 21 2022 *)

my(x='x+O('x^80)); concat([0,0,0], Vec(x^3/((1-x^2)*(1-x^3)^2))) \\ G. C. Greubel, Apr 03 2019

(x^3/((1-x^2)*(1-x^3)^2)).series(x, 80).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

a(n+2) = A321202(n) - A114209(n+1).
a(3n+1) = A002620(n+2).
a(3n+2) = A002620(n+1).
a(3n+3) = A002620(n+2).
G.f.: x^3/((1+x)*(1+x+x^2)^2*(1-x)^3). - Alois P. Heinz, Mar 19 2019
a(n) = a(n-2) + 2*a(n-3) - 2*a(n-5) - a(n-6) + a(n-8). - G. C. Greubel, Apr 03 2019
a(n) = (6*n*(2 + n) + 8*(4 + 3*n)*cos(2*n*Pi/3) - 8*sqrt(3)*n*sin(2*n*Pi/3) - 5 - 27*(-1)^n)/216. - Stefano Spezia, Apr 21 2022
From Ridouane Oudra, Nov 24 2024: (Start)
a(n) = (7*n/2 - 7*n^2/2 - 9*floor(n/2) + (6*n+4)*floor(2*n/3) + 4*floor(n/3))/18.
a(n) = A008133(n) - A069905(n-1).
a(n) = A002620(A008611(n)). (End)

More terms from Alois P. Heinz, Mar 19 2019

A376780 Triangular table read by rows: T(n,k) is the minimum number of minimal forbidden subgraphs of a graph with n vertices and k edges, n >= 1, 0 <= k <= n*(n-1)/2.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 3, 2, 3, 2, 2, 1, 2, 3, 2, 3, 3, 4, 3, 3, 2, 2, 1, 2, 3, 3, 2, 4, 3, 4, 5, 5, 4, 5, 5, 4, 2, 2, 1, 2, 3, 3, 2, 4, 4, 3, 4, 5, 4, 5, 5, 7, 6, 6, 5, 5, 4, 4, 2, 2, 1, 2, 3, 3, 3, 2, 4, 4, 3, 5, 6, 5, 6, 5, 5, 7, 6, 7, 10, 9, 9, 9, 8, 10, 5, 5, 4, 2, 2, 1

Offset: 1

Max Alekseyev, Oct 03 2024

			Table starts with
n = 1: 1
n = 2: 2, 1
n = 3: 2, 2, 2, 1
n = 4: 2, 3, 2, 3, 2, 2, 1
...

Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, 2024.

Cf. A002865, A006897, A103221, A371161, A371162, A376781, A376782.

A376781 Triangular table read by rows: T(n,k) is the maximum number of minimal forbidden subgraphs of a graph with n vertices and k edges, n >= 1, 0 <= k <= n*(n-1)/2.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 2, 3, 4, 4, 3, 2, 1, 2, 3, 4, 5, 5, 5, 5, 4, 3, 2, 1, 2, 3, 4, 6, 8, 8, 9, 13, 11, 11, 9, 6, 5, 3, 2, 1, 2, 3, 4, 6, 8, 8, 11, 16, 20, 23, 28, 31, 30, 33, 24, 22, 15, 10, 7, 3, 2, 1, 2, 3, 4, 6, 8, 9, 14, 21, 24, 31, 41, 57, 78, 86, 106, 123, 134, 149, 143, 138, 133, 75, 46, 37, 18, 11, 3, 2, 1

Offset: 1

Max Alekseyev, Oct 03 2024

			Table starts with
n = 1: 1
n = 2: 2, 1
n = 3: 2, 2, 2, 1
n = 4: 2, 3, 4, 4, 3, 2, 1
...

Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, 2024.

Cf. A371162 (row maximums).

Cf. A002865, A006897, A103221, A371161, A376780, A376782.

A154950 Riordan array (1/(1-x^4), x(1+x)/(1+x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 2, 1, 1, -1, -1, 3, 1, 0, 2, -4, 0, 4, 1, 0, 2, 2, -8, 2, 5, 1, 0, -2, 8, -2, -12, 5, 6, 1, 1, -2, -2, 18, -12, -15, 9, 7, 1, 0, 3, -12, 8, 28, -29, -16, 14, 8, 1, 0, 3, 3, -32, 38, 31, -53, -14, 20, 9, 1

Offset: 0

Paul Barry, Jan 17 2009

Row sums are A008619. Diagonal sums are A103221. Equal to A154948 times inverse of A007318.

			Triangle begins
1,
0, 1,
0, 1, 1,
0, -1, 2, 1,
1, -1, -1, 3, 1,
0, 2, -4, 0, 4, 1,
0, 2, 2, -8, 2, 5, 1,
0, -2, 8, -2, -12, 5, 6, 1,
1, -2, -2, 18, -12, -15, 9, 7, 1

Triangle T(n,k)=sum{i=0..n, sum{j=0..n+1, C(n+1-j,i+1)*C(i-1,j)}*(-1)^(i-k)*C(i,k)}.

cp's OEIS Frontend

A140953 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^11)*(1-x^13)).

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A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A266779 Molien series for invariants of finite Coxeter group A_10.

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A266780 Molien series for invariants of finite Coxeter group A_11.

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A376782 Triangle read by rows: T(n,m) is the number of unlabeled graphs with n vertices having m minimum forbidden subgraphs, n >= 1, 1 <= m <= A371162(n).

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A081753 a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.

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PARI

PARI

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A307018 Total number of parts of size 3 in the partitions of n into parts of size 2 and 3.

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A140953 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)*(1-x^13)).