A008667 -id:A008667 - cp's OEIS frontend

A266779 Molien series for invariants of finite Coxeter group A_10.

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)).

A333925 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j=2..k+1} 1/(1 - x^j).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 0, 0, 1, 0, 1, 1, 2, 2, 3, 1, 1, 0, 1, 0, 1, 1, 2, 2, 3, 2, 2, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 4, 2, 1, 0, 1, 0, 1, 1, 2, 2, 4, 3, 5, 3, 2, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 6, 5, 5, 2, 1, 0

Offset: 0

Ilya Gutkovskiy, Apr 10 2020

A(n,k) is the number of partitions of n into parts 2, 3, ..., k and k + 1.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  0,  0,  0,  0,  0,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  0,  1,  1,  1,  1,  ...
  0,  1,  1,  2,  2,  2,  ...
  0,  0,  1,  1,  2,  2,  ...

David A. Corneth, Table of n, a(n) for n = 0..10010 (first 141 rows antidiagonals flattened)
Index entries for Molien series

Columns k=0..12 give A000007, A059841, A103221, A266755, A008667, A037145, A001996, A266776, A266777, A266778, A266779, A266780, A266781.
Main diagonal gives A002865.
Cf. A008284, A058398.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^j), {j, 2, k + 1}], {x, 0, n}]][i - n], {i, 0, 13}, {n, 0, i}] // Flatten

G.f. of column k: Product_{j=2..k+1} 1/(1 - x^j).

cp's OEIS Frontend

A266779 Molien series for invariants of finite Coxeter group A_10.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A266780 Molien series for invariants of finite Coxeter group A_11.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A333925 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j=2..k+1} 1/(1 - x^j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula