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 A268587 #43 Feb 23 2025 13:33:05
%S A268587 0,0,0,0,5,24,85,264,760,2080,5488,14080,35328,87040,211200,505856,
%T A268587 1198080,2809856,6533120,15073280,34537472,78643200,178061312,
%U A268587 401080320,899153920,2006974464,4461690880,9881780224,21810380800,47982837760,105243475968
%N A268587 Expansion of x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.
%C A268587 a(n) is the number of North-East lattice paths from (0,0) to (n,n) that have exactly three east steps below y = x - 1 and no east steps above y = x+1. Details can be found in Section 4.1 in Pan and Remmel's link.
%H A268587 Robert Israel, <a href="/A268587/b268587.txt">Table of n, a(n) for n = 0..3270</a>
%H A268587 Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%H A268587 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F A268587 G.f.: x^4*(5 - 16*x + 13*x^2)/(1 - 2*x)^4.
%F A268587 From _Colin Barker_, Feb 08 2016: (Start)
%F A268587 a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 6.
%F A268587 a(n) = 2^(n-7)*(n-3)*(n+4)*(n+11)/3 for n > 2. (End)
%F A268587 E.g.f.: (33 + 60*x + 39*x^2 + (-33 + 6*x + 15*x^2 + 2*x^3)*exp(2*x))/96. - _G. C. Greubel_, May 24 2019
%p A268587 F:= gfun:-rectoproc({16*a(n)-32*a(n+1)+24*a(n+2)-8*a(n+3)+a(n+4), a(0)=0, a(1)=0,a(2)=0,a(3)=0,a(4)=5,a(5)=24,a(6)=85},a(n),remember):
%p A268587 map(F, [$0..40]); # _Robert Israel_, Feb 07 2016
%t A268587 CoefficientList[Series[x^4 (5 -16x +13x^2)/(1-2x)^4, {x, 0, 40}], x] (* _Michael De Vlieger_, Feb 08 2016 *)
%t A268587 LinearRecurrence[{8,-24,32,-16},{0,0,0,0,5,24,85},40] (* _Harvey P. Dale_, Feb 22 2025 *)
%o A268587 (PARI) concat(vector(4), Vec(x^4*(5-16*x+13*x^2)/(1-2*x)^4 + O(x^40))) \\ _Colin Barker_, Feb 08 2016
%o A268587 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0,0,0] cat Coefficients(R!( x^4*(5-16*x+13*x^2)/(1-2*x)^4 )); // _G. C. Greubel_, May 24 2019
%o A268587 (Sage) (x^4*(5-16*x+13*x^2)/(1-2*x)^4).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%o A268587 (GAP) Concatenation([0,0,0,0], List([3..40], n-> 2^(n-7)*(n-3)*(n+4)*(n+11)/3 )); # _G. C. Greubel_, May 24 2019
%Y A268587 Cf. A268462, A268586.
%K A268587 nonn,easy
%O A268587 0,5
%A A268587 _Ran Pan_, Feb 07 2016
%E A268587 Typo in name and g.f. corrected by _Georg Fischer_, May 24 2019