cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

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Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

A003403 G.f.: (1 + x^3 + x^4 + ... + x^12 + x^15)/Product_{i=1..10} (1 - x^i).

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 18, 27, 41, 60, 87, 122, 172, 235, 320, 430, 572, 751, 982, 1268, 1629, 2074, 2625, 3297, 4123, 5118, 6324, 7771, 9506, 11567, 14023, 16917, 20335, 24343, 29039, 34510, 40885, 48265, 56811, 66661, 78001, 91001, 105901, 122902, 142291, 164329, 189347
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Enumerates certain triangular arrays of integers.

References

  • 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. (10.5).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003402, A003404, A003405, A029073, A256975, A256976, A256977.

Programs

  • Maple
    (1+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^15)/mul(1-x^i,i=1..10);
  • Mathematica
    CoefficientList[Series[(1+Total[x^Range[3,12] ]+x^15)/Product[1 - x^i, {i,10}], {x,0,50}],x] (* Harvey P. Dale, Jun 24 2018 *)

Extensions

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

A003404 Number of solid partitions of n supported on graph of cube.

Original entry on oeis.org

1, 1, 4, 7, 14, 23, 41, 63, 104, 152, 230, 327, 470, 647, 897, 1202, 1616, 2117, 2775, 3566, 4580, 5787, 7301, 9092, 11298, 13885, 17028, 20688, 25076, 30154, 36172, 43094, 51221, 60511, 71323, 83622, 97822, 113893, 132323, 153083
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373 (see Section 98).
  • 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. shown below appears on page 121. - N. J. A. Sloane, Apr 22 2015]
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003402, A003403, A003405, A029073, A256975, A256976, A256977.

Programs

  • Mathematica
    CoefficientList[Series[(1+2*q^2+2*q^3+3*q^4+3*q^5+5*q^6+4*q^7+8*q^8+ 4*q^9+ 5*q^10+ 3*q^11+3*q^12+2*q^13+2*q^14+q^16)/((1-q)*(1-q^2)*(1-q^3)*(1-q^4)* (1-q^5)*(1-q^6)*(1-q^7)*(1-q^8)),{q,0,40}],q] (* Harvey P. Dale, Mar 07 2012 *)
LinearRecurrence[{1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1},{1,1,4,7,14,23,41,63,104,152,230,327,470,647,897,1202,1616,2117,2775,3566,4580,5787,7301,9092,11298,13885,17028,20688,25076,30154,36172,43094,51221,60511,71323,83622},50] (* Harvey P. Dale, Jun 11 2022 *)

Formula

G.f.: (1 + 2*q^2 + 2*q^3 + 3*q^4 + 3*q^5 + 5*q^6 + 4*q^7 + 8*q^8 + 4*q^9 + 5*q^10 + 3*q^11 + 3*q^12 + 2*q^13 + 2*q^14 + q^16)/((1 - q)*(1 - q^2)*(1 - q^3)*(1 - q^4)*(1 - q^5)*(1 - q^6)*(1 - q^7)*(1 - q^8)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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