cp's OEIS Frontend

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

%I A003402 M1003 #42 Oct 21 2022 22:07:31
%S A003402 1,1,2,4,6,9,14,19,27,37,49,64,84,106,134,168,207,253,309,371,445,530,
%T A003402 626,736,863,1003,1163,1343,1543,1766,2017,2291,2597,2935,3305,3712,
%U A003402 4161,4647,5181,5763,6394,7079,7825,8627,9497,10436,11445,12531,13702,14952
%N A003402 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)).
%C A003402 Enumerates certain triangular arrays of integers.
%C A003402 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
%D A003402 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
%D A003402 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003402 Robert Israel, <a href="/A003402/b003402.txt">Table of n, a(n) for n = 0..10000</a>
%H A003402 J. C. P. Miller, <a href="http://dx.doi.org/10.1017/CBO9780511662072.018">On the enumeration of partially ordered sets of integers</a>, 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).
%H A003402 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A003402 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, -1, -2, -1, -1, 2, 2, 2, -1, -1, -2, -1, 1, 1, 1, -1).
%F A003402 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
%F A003402 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
%p A003402 A079978:= n -> `if`(n mod 3 = 0, 1, 0):
%p A003402 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):
%p A003402 seq(F(n), n= 0..100); # _Robert Israel_, Apr 22 2015
%t A003402 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 *)
%o A003402 (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
%Y A003402 Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.
%Y A003402 Cf. A003403, A003404, A003405, A029073, A256975, A256976, A256977.
%K A003402 nonn,easy
%O A003402 0,3
%A A003402 _N. J. A. Sloane_
%E A003402 Entry revised by _N. J. A. Sloane_, Apr 22 2015