A266769 - OEIS

%I #33 Sep 08 2022 08:46:15

%S 1,1,3,4,7,9,14,17,24,29,38,45,57,66,81,93,111,126,148,166,192,214,

%T 244,270,305,335,375,410,455,495,546,591,648,699,762,819,889,952,1029,

%U 1099,1183,1260,1352,1436,1536,1628,1736,1836,1953,2061

%N Expansion of 1/((1-x)*(1-x^2)^2*(1-x^3)).

%C This is the same as A008763 but without the four leading zeros. There are so many situations where one wants this sequence rather than A008763 that it seems appropriate for it to have its own entry.

%C But see A008763 (still the main entry) for numerous applications and references.

%C Also, Molien series for invariants of finite Coxeter group D_4 (bisected).

%C 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.

%C Euler transform of length 3 sequence [1, 2, 1]. - _Michael Somos_, Jun 26 2017

%C a(n) is the number of partitions of n into parts 1, 2, and 3, where there are two sorts of parts 2. - _Joerg Arndt_, Jun 27 2017

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

%H G. C. Greubel, <a href="/A266769/b266769.txt">Table of n, a(n) for n = 0..1000</a>

%H L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,-1,2,1,-1).

%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) for n>7. - _Vincenzo Librandi_, Jan 11 2016

%F a(n) = -a(-8-n) for all n in Z. - _Michael Somos_, Jun 26 2017

%t CoefficientList[Series[1/((1-x)*(1-x^2)^2*(1-x^3)), {x, 0, 50}], x] (* _JungHwan Min_, Jan 10 2016 *)

%t LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1}, {1, 1, 3, 4, 7, 9, 14, 17}, 100] (* _Vincenzo Librandi_, Jan 11 2016 *)

%o (PARI) Vec(1/((1-x)*(1-x^2)^2*(1-x^3)) + O(x^100)) \\ _Michel Marcus_, Jan 11 2016

%o (Magma) I:=[1,1,3,4,7,9,14,17]; [n le 8 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-3)-2*Self(n-4)-Self(n-5)+2*Self(n-6)+Self(n-7)-Self(n-8): n in [1..60]]; // _Vincenzo Librandi_, Jan 11 2016

%o (PARI) {a(n) = (9*(n+4)*(-1)^n + 2*n^3 + 24*n^2 + 87*n + 157) \ 144}; /* _Michael Somos_, Jun 26 2017 */

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

%Y A variant of A008763.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Jan 10 2016