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 A382540 #16 May 12 2025 13:59:42
%S A382540 1,3,12,52,231,1035,4650,20898,93849,420935,1885248,8430588,37642819,
%T A382540 167824905,747143298,3321632498,14747814597,65397373761,289652172896,
%U A382540 1281454446408,5663228541975,25002457308487,110275917725658,485935158536874,2139412626785505
%N A382540 Expansion of 1/(1 - x/(1 - 4*x)^(1/2))^3.
%H A382540 Vincenzo Librandi, <a href="/A382540/b382540.txt">Table of n, a(n) for n = 0..600</a>
%F A382540 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n-k/2-1,n-k).
%F A382540 D-finite with recurrence 3*(-n+1)*a(n) +6*(6*n-11)*a(n-1) +2*(-71*n+199)*a(n-2) +4*(44*n-183)*a(n-3) +3*(11*n+5)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - _R. J. Mathar_, Apr 02 2025
%t A382540 Table[Sum[(4)^(n-k)* Binomial[k+2,2]*Binomial[n-k/2-1, n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, May 12 2025 *)
%o A382540 (PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(n-k/2-1, n-k));
%o A382540 (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^3)); // _Vincenzo Librandi_, May 12 2025
%Y A382540 Cf. A026671, A382539.
%Y A382540 Cf. A382542.
%K A382540 nonn,easy
%O A382540 0,2
%A A382540 _Seiichi Manyama_, Mar 31 2025