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%I #31 Apr 16 2023 12:34:49
%S 1,4,11,26,55,109,208,389,722,1339,2488,4634,8646,16146,30160,56333,
%T 105198,196413,366672,684475,1277701,2385080,4452277,8311254,15515091,
%U 28963012,54067156,100930660,188413624,351723304,656583197
%N Expansion of x^6*(1 + x^3)/(1 - 4*x + 5*x^2 - x^3 - 2*x^4 + x^6 + x^7 - 2*x^8 + x^9).
%C This corresponds to S(213,1,x) of Langley if one uses Theorem 8. Note that all three expressions for S(213;t,x), S(213;1,x) and the series on page 22 are mutually incompatible, so we show the sequence one would most likely see in other publications.
%H Vincenzo Librandi, <a href="/A290989/b290989.txt">Table of n, a(n) for n = 6..1000</a>
%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order</a>, J. Int. Seq. 14 (2011) # 11.4.2.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,0,-1,-1,2,-1).
%F G.f.: x^6*(1 + x)*(1 - x + x^2)/((1 - x)*(1 - 2*x + x^3 - x^4)*(1 - x + x^4)).
%F a(n) = -2 + (1/19)*( 9*A099530(n+1) + 15*A099530(n) + 2*A099530(n-1) - A099530(n- 2) + 10*A059633(n+4) - 6*A059633(n+3) - 16*A059633(n+2) - A059633(n+1) ). - _G. C. Greubel_, Apr 12 2023
%t DeleteCases[#, 0] &@ CoefficientList[Series[x^6*(1+x^3)/(1 -4x +5x^2 -x^3 -2x^4 +x^6 +x^7 -2x^8 +x^9), {x, 0, 36}], x] (* _Michael De Vlieger_, Aug 16 2017 *)
%t LinearRecurrence[{4,-5,1,2,0,-1,-1,2,-1}, {1,4,11,26,55,109,208,389,722}, 80] (* _Vincenzo Librandi_, Aug 17 2017 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^6*(1+x^3)/((1-x)*(1-2*x+x^3-x^4)*(1-x+x^4)) )); // _G. C. Greubel_, Apr 12 2023
%o (SageMath)
%o def A290989_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^6*(1+x^3)/((1-x)*(1-x+x^4)*(1-2*x+x^3-x^4)) ).list()
%o a=A290989_list(50); a[6:] # _G. C. Greubel_, Apr 12 2023
%Y Cf. A059633, A099530, A290986, A290987.
%K nonn,easy
%O 6,2
%A _R. J. Mathar_, Aug 16 2017
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