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%I #16 Jun 24 2023 09:20:47
%S 1,1,1,2,3,5,9,15,26,45,78,134,232,402,695,1205,2086,3613,6259,10841,
%T 18780,32531,56354,97621,169111,292954,507488,879136,1522947,2638242,
%U 4570298,7917253,13715281,23759370,41159039,71300984,123516755,213971647,370669282
%N Expansion of g.f. 1/ (1-x^1*(1-x^(m+1))/ (1-x^2*(1-x^(m+2))/ (1- ... ))) for m=9.
%H Alois P. Heinz, <a href="/A228645/b228645.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul D. Hanna et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2013-July/011445.html">Formula Needed for a Family of Continued Fractions</a> and follow-up messages on the SeqFan list, Jul 28 2013
%H <a href="/index/Rec#order_35">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, 0, 0, -1, -1, -2, -1, -3, -2, 0, 1, 3, 4, 4, 4, 4, 2, 0, -2, -3, -5, -4, -4, -3, -2, 0, 1, 1, 2, 2, 1, 1, 1).
%F G.f.: -(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)).
%p a:= n-> coeff(series(-(x^30 +x^29 +x^28 +2*x^27 +2*x^26 +2*x^25 +x^24 +x^23 -x^22 -2*x^21 -2*x^20 -4*x^19 -4*x^18 -3*x^17 -2*x^16 -x^15 +2*x^13 +2*x^12 +3*x^11 +3*x^10 +x^9 +x^8 -x^5 -x^4 -x^3 -x^2+1) / ((x-1)*(x^2 +x+1)*(x^6 +x^3+1)*(x^26 +x^25 +x^24 +2*x^23 +2*x^22 +x^21 +x^20 -2*x^18 -2*x^17 -3*x^16 -3*x^15 -3*x^14 -x^13 -x^12 +x^11 +2*x^10 +2*x^9 +2*x^8 +x^7 +x^6 -x^3 -x^2 -x+1)), x, n+1), x, n): seq(a(n), n=0..50);
%t nMax = 39; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x] &; A228645 = col[9][[1 ;; nMax]] (* _Jean-François Alcover_, Nov 03 2016 *)
%Y Cf. A143064 (m=0), A227360 (m=2), A173173 (m=3), A227374 (m=4), A227375 (m=5), A228646 (m=6), A228644 (m=7).
%Y Column m=9 of A185646.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Aug 28 2013
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