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%I #25 Sep 08 2022 08:46:13
%S 1,1,2,4,7,13,23,41,74,132,236,422,754,1348,2409,4305,7694,13750,
%T 24573,43915,78481,140255,250652,447944,800528,1430636,2556712,
%U 4569140,8165581,14592837,26079086,46606340,83290915,148850489,266013023,475396009,849587598,1518311204,2713397556,4849154954,8666000202
%N Expansion of 1/(1 - x - x^2 - x^3 + x^6 + x^7).
%C This sequence counts the partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs (1,3);(3,4);(2,4) is unimportant. Alternatively the order of complementary pairs (1,2);(1,4);(2,3) is important.
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,0,0,-1,-1).
%F a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-6) - a(n-7).
%F G.f.: 1/((1 - x)*(1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6)).
%e a(7)=41; the corresponding partitions (cf. comment) are: (43), (241=421), (124=142), (412), (214), (4111), (1411), (1141), (1114), (331=313=133), (322), (232), (223), (3112=1312=1132), (2113=2131=2311), (1213=1231), (3121=1321), (3211), (1123), (31111=13111=11311=11131=11113), (2221)=four, (22111)=ten, (211111)=six, (1111111).
%t CoefficientList[Series[1/(1 - x - x^2 - x^3 + x^6 + x^7), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 07 2015 *)
%t LinearRecurrence[{1,1,1,0,0,-1,-1},{1,1,2,4,7,13,23},50] (* _Harvey P. Dale_, Aug 21 2021 *)
%o (PARI) Vec(1/(1 - x - x^2 - x^3 + x^6 + x^7) + O(x^50)) \\ _Michel Marcus_, Aug 06 2015
%o (Magma) I:=[1,1,2,4,7,13,23]; [n le 7 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) - Self(n-6) - Self(n-7): n in [1..45]]; // _Vincenzo Librandi_, Aug 07 2015
%Y Cf. A023435, A080239, A260710.
%K nonn,easy
%O 0,3
%A _David Neil McGrath_, Aug 04 2015
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