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 A382616 #18 Apr 02 2025 15:57:37
%S A382616 1,2,9,34,124,444,1567,5466,18885,64732,220403,746166,2513678,8431650,
%T A382616 28175256,93834240,311565255,1031723268,3408137644,11233323692,
%U A382616 36950587185,121319416734,397649266199,1301332828086,4252515425757,13877722224278,45232020345642
%N A382616 Expansion of 1/(1 - x/(1 - x)^3)^2.
%H A382616 Vincenzo Librandi, <a href="/A382616/b382616.txt">Table of n, a(n) for n = 0..400</a>
%H A382616 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,26,-17,6,-1).
%F A382616 a(n) = Sum_{k=0..n} (k+1) * binomial(n+2*k-1,n-k).
%F A382616 a(n) = 8*a(n-1) - 22*a(n-2) + 26*a(n-3) - 17*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.
%F A382616 G.f.: (x-1)^6/(x^3-3*x^2+4*x-1)^2.
%t A382616 Table[Sum[(k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,26}] (* _Vincenzo Librandi_, Apr 02 2025 *)
%o A382616 (PARI) a(n) = sum(k=0, n, (k+1)*binomial(n+2*k-1, n-k));
%o A382616 (Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x/(1 - x)^3)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // _Vincenzo Librandi_, Apr 02 2025
%Y A382616 Cf. A045623, A290917.
%Y A382616 Cf. A052529, A382615.
%K A382616 nonn,easy
%O A382616 0,2
%A A382616 _Seiichi Manyama_, Mar 31 2025