A266755 -id:A266755 - cp's OEIS frontend

A266780 Molien series for invariants of finite Coxeter group A_11.

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)).

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

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

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

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

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

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

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)).

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).

A266771 Molien series for invariants of finite Coxeter group D_8 (bisected).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 27, 36, 51, 67, 92, 118, 156, 198, 256, 319, 404, 498, 620, 755, 926, 1116, 1353, 1615, 1935, 2291, 2720, 3194, 3759, 4384, 5120, 5932, 6879, 7923, 9131, 10458, 11981, 13654, 15561, 17648, 20014, 22600, 25514, 28692, 32255, 36134, 40464, 45167

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.

Index entries for Molien series

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

Take[CoefficientList[Series[1/((1-x^8)Times@@(1-x^Range[2,14,2])),{x,0,100}],x],{1,-1,2}] (* Harvey P. Dale, Jan 02 2018 *)

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

cp's OEIS Frontend

A266780 Molien series for invariants of finite Coxeter group A_11.

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A266772 Molien series for invariants of finite Coxeter group D_9.

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

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A266774 Molien series for invariants of finite Coxeter group D_11.

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Magma

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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).

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A266771 Molien series for invariants of finite Coxeter group D_8 (bisected).

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