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 A382613 #17 Apr 08 2025 13:09:24
%S A382613 1,2,9,28,88,270,808,2386,6960,20104,57607,163950,463907,1306104,
%T A382613 3661248,10223820,28452400,78941412,218426608,602886704,1660329597,
%U A382613 4563175466,12517834605,34280427828,93729509848,255900484218,697712467704,1899912606358,5167488465184
%N A382613 Expansion of 1/(1 - x*(1 + x)^3)^2.
%H A382613 Vincenzo Librandi, <a href="/A382613/b382613.txt">Table of n, a(n) for n = 0..500</a>
%H A382613 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,0,-13,-20,-15,-6,-1).
%F A382613 a(n) = Sum_{k=0..n} (k+1) * binomial(3*k,n-k).
%F A382613 a(n) = 2*a(n-1) + 5*a(n-2) - 13*a(n-4) - 20*a(n-5) - 15*a(n-6) - 6*a(n-7) - a(n-8).
%F A382613 G.f.: 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2. - _Vincenzo Librandi_, Apr 08 2025
%t A382613 Table[Sum[(k+1)*Binomial[3*k,n-k],{k,0,n}],{n,0,28}] (* _Vincenzo Librandi_, Apr 08 2025 *)
%o A382613 (PARI) a(n) = sum(k=0, n, (k+1)*binomial(3*k, n-k));
%o A382613 (Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x - 3*x^2 - 3*x^3 - x^4)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // _Vincenzo Librandi_, Apr 08 2025
%Y A382613 Cf. A099234, A382614.
%Y A382613 Cf. A001629, A362126.
%K A382613 nonn
%O A382613 0,2
%A A382613 _Seiichi Manyama_, Mar 31 2025