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A029011 Expansion of 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^6)).

%I A029011 #24 Jun 26 2025 08:01:37
%S A029011 1,1,2,2,3,4,6,7,9,10,13,15,19,21,25,28,33,37,43,47,54,59,67,73,82,89,
%T A029011 99,107,118,127,140,150,164,175,190,203,220,234,252,267,287,304,326,
%U A029011 344,367,387,412,434,461,484,513,538,569,596,629,658,693,724,761,794
%N A029011 Expansion of 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^6)).
%C A029011 Number of partitions of n into parts 1, 2, 5 and 6. - _David Neil McGrath_, Dec 06 2014
%C A029011 Number of walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 5-loop, 6-loop) where the order of loops is unimportant. - _David Neil McGrath_, Dec 06 2014
%H A029011 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A029011 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,1,0,-2,0,1,0,-1,1,1,-1).
%F A029011 G.f.: 1/((1-x)(1-x^2)(1-x^5)(1-x^6)).
%F A029011 a(n) = -a(-14-n).
%F A029011 a(n) = a(n-2) + a(n-5) + a(n-6) - a(n-7) - a(n-8) - a(n-11) + a(n-13) + 1.
%F A029011 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-5) - 2*a(n-7) + a(n-9) - a(n-11) + a(n-12) + a(n-13) - a(n-14). - _David Neil McGrath_, Dec 06 2014
%F A029011 a(n) = 1 + [(n mod 30)=12] + floor((2n^3 + 42*n^2 + (261+15*(-1)^n)*n)/720). - _Hoang Xuan Thanh_, Jun 25 2025
%e A029011 There are 6 partitions of n=6 into parts 1, 2, 5 and 6. These are (6)(51)(222)(2211)(21111)(111111). - _David Neil McGrath_, Dec 06 2014
%t A029011 CoefficientList[Series[1/((1-x)(1-x^2)(1-x^5)(1-x^6)),{x,0,60}],x]
%t A029011 LinearRecurrence[{1,1,-1,0,1,0,-2,0,1,0,-1,1,1,-1},{1,1,2,2,3,4,6,7,9,10,13,15,19,21},70] (* _Harvey P. Dale_, Dec 14 2020 *)
%o A029011 (PARI) a(n)=if(n<-13,-a(-14-n),polcoeff(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^6))+x*O(x^n),n))
%Y A029011 Cf. A029177(2*n) = a(n).
%K A029011 nonn,easy
%O A029011 0,3
%A A029011 _N. J. A. Sloane_