This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A260710 #29 Sep 26 2023 01:59:25
%S A260710 1,1,2,3,6,9,16,25,43,69,116,188,313,511,846,1386,2288,3756,6191,
%T A260710 10174,16756,27552,45357,74604,122787,201996,332414,546901,899946,
%U A260710 1480699,2436459,4008858,6596366,10853563,17858788,29384804,48350401,79555943,130902711
%N A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).
%C A260710 This sequence counts partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs of numbers (1,2);(2,3);(3,4) is unimportant. Alternatively the order of the complementary pairs (1,4);(1,3);(2,4) is important.
%H A260710 <a href="/index/Par#partN">Index entries for related partition-counting sequences </a>
%H A260710 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,1,-1,0,-1).
%F A260710 G.f: 1/(1 - x - x^2 - x^4 + x^5 + x^7).
%F A260710 a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-5) - a(n-7).
%F A260710 Construct the matrix array T(n,j) = [A^*j]*[S^*(j-1)] where A=(1,1,0,1,-1,0,-1) and S=(0,1,0,...) (A063524). [* is convolution operation] Define S^*0=I with I=(1,0,...). a(n) = Sum_{j=1..n} T(n,j).
%e A260710 There are 25 partially ordered partitions of 7, i.e., a(7) = 25. These are (43=34),(421=412),(124=214),(241),(142),(4111),(1411),(1141),(1114),(331),(313),(133),(1132=1123),(2131=1231),(1312=1321),(2311=3211),(31111),(13111),(11311),(11131),(11113),(2221=four),(22111=ten),(211111=six),(1111111).
%t A260710 LinearRecurrence[{1, 1, 0, 1, -1, 0, -1}, {1, 1, 2, 3, 6, 9, 16}, 50] (* _Vincenzo Librandi_, Aug 04 2015 *)
%o A260710 (Magma) I:=[1,1,2,3,6,9,16]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-5)-Self(n-7): n in [1..40]]; // _Vincenzo Librandi_, Aug 04 2015
%o A260710 (PARI) Vec(1/(1 - x - x^2 - x^4 + x^5 + x^7) + O(x^50)) \\ _Michel Marcus_, Aug 06 2015
%Y A260710 Cf. A023435, A080239, A023434, A254685, A116732, A004695.
%K A260710 nonn,easy
%O A260710 0,3
%A A260710 _David Neil McGrath_, Jul 30 2015