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 A382495 #16 May 12 2025 10:13:57
%S A382495 1,0,4,4,10,60,30,300,335,1000,2506,3500,11879,17304,44220,88592,
%T A382495 161865,385704,660964,1475100,2807956,5459860,11313094,20816004,
%U A382495 42774780,80798128,157292750,307887904,579776799,1138007940,2146348214,4126143900,7878910238,14878269368
%N A382495 a(n) = Sum_{k=0..floor(n/2)} binomial(k+3,3) * binomial(2*k,2*n-4*k).
%H A382495 Vincenzo Librandi, <a href="/A382495/b382495.txt">Table of n, a(n) for n = 0..1500</a>
%H A382495 <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,8,8,-28,-40,28,72,2,16,20,-80,-114,56,-68,0,35,40,-96,128,-110,64,-28,8,-1).
%F A382495 G.f.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x^2-x^3)^(4-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^4.
%F A382495 a(n) = 8*a(n-2) + 8*a(n-3) - 28*a(n-4) - 40*a(n-5) + 28*a(n-6) + 72*a(n-7) + 2*a(n-8) + 16*a(n-9) + 20*a(n-10) - 80*a(n-11) - 114*a(n-12) + 56*a(n-13) - 68*a(n-14) + 35*a(n-16) + 40*a(n-17) - 96*a(n-18) + 128*a(n-19) - 110*a(n-20) + 64*a(n-21) - 28*a(n-22) + 8*a(n-23) - a(n-24).
%t A382495 Table[Sum[Binomial[k+3,3]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,33}] (* _Vincenzo Librandi_, May 12 2025 *)
%o A382495 (PARI) a(n) = sum(k=0, n\2, binomial(k+3, 3)*binomial(2*k, 2*n-4*k));
%o A382495 (PARI) my(N=3, 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 A382495 (Magma) [&+[Binomial(k+3, 3)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..40]]; // _Vincenzo Librandi_, May 12 2025
%Y A382495 Cf. A376729, A382300, A382494.
%Y A382495 Cf. A034839, A382470.
%K A382495 nonn,easy
%O A382495 0,3
%A A382495 _Seiichi Manyama_, Mar 29 2025