A266755 -id:A266755 - cp's OEIS frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

1, 3, 5, 9, 10, 15, 25, 27, 30, 45, 50, 75, 81, 90, 100, 125, 135, 150, 225, 243, 250, 270, 300, 375, 405, 450, 500, 625, 675, 729, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is 5-smooth if its prime divisors are all <= 5.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
       1: {}            125: {3,3,3}
       3: {2}           135: {2,2,2,3}
       5: {3}           150: {1,2,3,3}
       9: {2,2}         225: {2,2,3,3}
      10: {1,3}         243: {2,2,2,2,2}
      15: {2,3}         250: {1,3,3,3}
      25: {3,3}         270: {1,2,2,2,3}
      27: {2,2,2}       300: {1,1,2,3,3}
      30: {1,2,3}       375: {2,3,3,3}
      45: {2,2,3}       405: {2,2,2,2,3}
      50: {1,3,3}       450: {1,2,2,3,3}
      75: {2,3,3}       500: {1,1,3,3,3}
      81: {2,2,2,2}     625: {3,3,3,3}
      90: {1,2,2,3}     675: {2,2,2,3,3}
     100: {1,1,3,3}     729: {2,2,2,2,2,2}

Allowing any number of parts and sum gives A051037, counted by A001399.
These are Heinz numbers of the partitions counted by A266755.
Allowing parts > 5 gives A344291, counted by A110618.
The non-3-smooth case is A344294, counted by A325691.
Requiring the sum of prime indices to be even gives A344295.
A000070 counts non-multigraphical partitions, ranked by A344292.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A300061 ranks partitions of even numbers, with 5-smooth case A344297.
Cf. A000041, A000244, A026811, A080193, A244990, A261144, A279622.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]<=5&]

Intersection of A051037 and A344291.

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

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 (0,1,0,1,0,0,1,0,-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

A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

Original entry on oeis.org

5, 10, 15, 25, 30, 45, 50, 75, 90, 100, 125, 135, 150, 225, 250, 270, 300, 375, 405, 450, 500, 625, 675, 750, 810, 900, 1000, 1125, 1215, 1250, 1350, 1500, 1875, 2025, 2250, 2430, 2500, 2700, 3000, 3125, 3375, 3645, 3750, 4050, 4500, 5000, 5625, 6075, 6250

Offset: 1

Gus Wiseman, May 16 2021

nonn

A number is d-smooth iff its prime divisors are all <= d.

A prime index of k is a number m such that prime(m) divides k, and the multiset of prime indices of k is row k of A112798. This row has length A001222(k) and sum A056239(k).

			The sequence of terms together with their prime indices begins:
       5: {3}           270: {1,2,2,2,3}
      10: {1,3}         300: {1,1,2,3,3}
      15: {2,3}         375: {2,3,3,3}
      25: {3,3}         405: {2,2,2,2,3}
      30: {1,2,3}       450: {1,2,2,3,3}
      45: {2,2,3}       500: {1,1,3,3,3}
      50: {1,3,3}       625: {3,3,3,3}
      75: {2,3,3}       675: {2,2,2,3,3}
      90: {1,2,2,3}     750: {1,2,3,3,3}
     100: {1,1,3,3}     810: {1,2,2,2,2,3}
     125: {3,3,3}       900: {1,1,2,2,3,3}
     135: {2,2,2,3}    1000: {1,1,1,3,3,3}
     150: {1,2,3,3}    1125: {2,2,3,3,3}
     225: {2,2,3,3}    1215: {2,2,2,2,2,3}
     250: {1,3,3,3}    1250: {1,3,3,3,3}

Allowing any number of parts and sum gives A080193, counted by A069905.
The partitions with these Heinz numbers are counted by A325691.
Relaxing the smoothness conditions gives A344291, counted by A110618.
Allowing 3-smoothness gives A344293, counted by A266755.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n whose length is n/2, ranked by A340387.
A051037 lists 5-smooth numbers (complement: A279622).
A056239 adds up prime indices, row sums of A112798.
A257993 gives the least gap of the partition with Heinz number n.
A300061 lists numbers with even sum of prime indices (5-smooth: A344297).
A342050/A342051 list Heinz numbers of partitions with even/odd least gap.
Cf. A000041, A001399, A002865, A027336, A244990, A261144, A344295.

Select[Range[1000],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&&Max@@First/@FactorInteger[#]==5&]

Intersection of A080193 and A344291.

A344297 Heinz numbers of integer partitions of even numbers with no part greater than 3.

Original entry on oeis.org

1, 3, 4, 9, 10, 12, 16, 25, 27, 30, 36, 40, 48, 64, 75, 81, 90, 100, 108, 120, 144, 160, 192, 225, 243, 250, 256, 270, 300, 324, 360, 400, 432, 480, 576, 625, 640, 675, 729, 750, 768, 810, 900, 972, 1000, 1024, 1080, 1200, 1296, 1440, 1600, 1728, 1875, 1920

Offset: 1

Gus Wiseman, May 16 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
       1: {}                 81: {2,2,2,2}
       3: {2}                90: {1,2,2,3}
       4: {1,1}             100: {1,1,3,3}
       9: {2,2}             108: {1,1,2,2,2}
      10: {1,3}             120: {1,1,1,2,3}
      12: {1,1,2}           144: {1,1,1,1,2,2}
      16: {1,1,1,1}         160: {1,1,1,1,1,3}
      25: {3,3}             192: {1,1,1,1,1,1,2}
      27: {2,2,2}           225: {2,2,3,3}
      30: {1,2,3}           243: {2,2,2,2,2}
      36: {1,1,2,2}         250: {1,3,3,3}
      40: {1,1,1,3}         256: {1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}       270: {1,2,2,2,3}
      64: {1,1,1,1,1,1}     300: {1,1,2,3,3}
      75: {2,3,3}           324: {1,1,2,2,2,2}

These partitions are counted by A007980.
Including partitions of odd numbers gives A051037 (complement: A279622).
Allowing parts > 3 gives A300061.
A001358 lists semiprimes.
A035363 counts partitions whose length is half their sum, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
Cf. A001399, A002620, A030229, A080193, A244990, A261144, A266755, A344291, A344293, A344294.

Select[Range[1000],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&&Max@@First/@FactorInteger[#]<=Prime[3]&]

Intersection of A051037 and A300061.

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)).

A344295 Heinz numbers of partitions of 2*n with at most n parts, none greater than 3, for some n.

Original entry on oeis.org

1, 3, 9, 10, 25, 27, 30, 75, 81, 90, 100, 225, 243, 250, 270, 300, 625, 675, 729, 750, 810, 900, 1000, 1875, 2025, 2187, 2250, 2430, 2500, 2700, 3000, 5625, 6075, 6250, 6561, 6750, 7290, 7500, 8100, 9000, 10000, 15625, 16875, 18225, 18750, 19683, 20250, 21870

Offset: 1

Gus Wiseman, May 15 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}
      3: {2}
      9: {2,2}
     10: {1,3}
     25: {3,3}
     27: {2,2,2}
     30: {1,2,3}
     75: {2,3,3}
     81: {2,2,2,2}
     90: {1,2,2,3}
    100: {1,1,3,3}
    225: {2,2,3,3}
    243: {2,2,2,2,2}
    250: {1,3,3,3}
    270: {1,2,2,2,3}
    300: {1,1,2,3,3}

These partitions are counted by A001399.
Allowing any number of parts and sum gives A051037.
Allowing parts > 3 and any length gives A300061.
Not requiring the sum of prime indices to be even gives A344293.
Allowing any number of parts (but still with even sum) gives A344297.
Allowing parts > 3 gives A344413.
A001358 lists semiprimes.
A025065 counts partitions of n with at least n/2 parts, ranked by A344296.
A035363 counts partitions of n of length n/2, ranked by A340387.
A056239 adds up prime indices, row sums of A112798.
A110618 counts partitions of n with at most n/2 parts, ranked by A344291.
A344414 counts partitions of n with all parts >= n/2, ranked by A344296.
Cf. A030229, A080193, A244990, A261144, A266755, A279622, A334433, A344294.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],EvenQ[Total[primeMS[#]]]&&PrimeOmega[#]<=Total[primeMS[#]]/2&&Max@@primeMS[#]<=3&]

Intersection of A300061 (even Heinz weight), A344291 (Omega > half Heinz weight), and A051037 (5-smooth).

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)).

cp's OEIS Frontend

A344293 5-smooth numbers n whose sum of prime indices A056239(n) is at least twice the number of prime indices A001222(n).

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

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A344294 5-smooth but not 3-smooth numbers k such that A056239(k) >= 2*A001222(k).

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A344297 Heinz numbers of integer partitions of even numbers with no part greater than 3.

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A266777 Molien series for invariants of finite Coxeter group A_8.

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