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A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).

%I A299337 #13 Dec 19 2024 12:26:30
%S A299337 1,2,8,14,35,56,112,168,294,420,672,924,1386,1848,2640,3432,4719,6006,
%T A299337 8008,10010,13013,16016,20384,24752,30940,37128,45696,54264,65892,
%U A299337 77520,93024,108528,128877,149226,175560,201894,235543,269192,311696,354200,407330
%N A299337 Expansion of 1 / ((1 - x)^7*(1 + x)^5).
%H A299337 Michael De Vlieger, <a href="/A299337/b299337.txt">Table of n, a(n) for n = 0..10000</a>
%H A299337 Brian Hopkins and Aram Tangboonduangjit, <a href="https://arxiv.org/abs/2412.11528">Water Cells in Compositions of 1s and 2s</a>, arXiv:2412.11528 [math.CO], 2024. See p. 3.
%H A299337 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).
%F A299337 a(n) = (2*n^6 + 72*n^5 + 1040*n^4 + 7680*n^3 + 30368*n^2 + 60288*n + 46080) / 46080 for n even.
%F A299337 a(n) = (2*n^6 + 72*n^5 + 1010*n^4 + 6960*n^3 + 24278*n^2 + 39048*n + 20790) / 46080 for n odd.
%F A299337 a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>11.
%t A299337 LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {1, 2, 8, 14, 35, 56, 112, 168, 294, 420, 672, 924}, 41] (* _Michael De Vlieger_, Dec 19 2024 *)
%o A299337 (PARI) Vec(1 / ((1 - x)^7*(1 + x)^5) + O(x^40))
%Y A299337 Cf. A001769, A060099, A299335, A299336, A299338.
%K A299337 nonn,easy
%O A299337 0,2
%A A299337 _Colin Barker_, Feb 07 2018