A003402 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)).

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• Robert Israel, Table of n, a(n) for n = 0..10000
• 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. (8.1).
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, -1, -2, -1, -1, 2, 2, 2, -1, -1, -2, -1, 1, 1, 1, -1).

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