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 A382300 #23 May 12 2025 13:59:48
%S A382300 1,0,2,2,3,18,7,60,65,144,356,410,1272,1722,3743,7202,11482,25566,
%T A382300 40421,81610,147169,259810,507267,867792,1659112,2961860,5362592,
%U A382300 9940420,17583485,32564548,58228386,105606458,191831767,343313042,625086891,1119760040,2023087045
%N A382300 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(2*k,2*n-4*k).
%H A382300 Vincenzo Librandi, <a href="/A382300/b382300.txt">Table of n, a(n) for n = 0..1500</a>
%H A382300 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,4,-6,-4,-2,-4,-5,8,-6,4,-1).
%F A382300 G.f.: ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2.
%F A382300 a(n) = 4*a(n-2) + 4*a(n-3) - 6*a(n-4) - 4*a(n-5) - 2*a(n-6) - 4*a(n-7) - 5*a(n-8) + 8*a(n-9) - 6*a(n-10) + 4*a(n-11) - a(n-12).
%t A382300 Table[Sum[(k+1)*Binomial[2*k,2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* _Vincenzo Librandi_, May 11 2025 *)
%o A382300 (PARI) a(n) = sum(k=0, n\2, (k+1)*binomial(2*k, 2*n-4*k));
%o A382300 (PARI) my(N=1, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
%o A382300 (Magma) R<x>:=PowerSeriesRing(Rationals(), 37); Coefficients(R!( ((1-x^2-x^3)^2 + 4*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^2)); // _Vincenzo Librandi_, May 11 2025
%Y A382300 Cf. A376729, A382494, A382495.
%Y A382300 Cf. A381421, A382496.
%Y A382300 Cf. A034839, A375218.
%K A382300 nonn,easy
%O A382300 0,3
%A A382300 _Seiichi Manyama_, Mar 29 2025