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 A377197 #25 May 11 2025 00:56:47
%S A377197 1,6,36,206,1146,6258,33728,180018,953628,5021698,26315676,137350746,
%T A377197 714455826,3705635646,19171860336,98973407550,509963556330,
%U A377197 2623133951730,13472299015580,69098721151530,353966981339070,1811212435206070,9258333786967920,47281424213258070
%N A377197 Expansion of 1/(1 - 4*x/(1-x))^(3/2).
%H A377197 Vincenzo Librandi, <a href="/A377197/b377197.txt">Table of n, a(n) for n = 0..1000</a>
%F A377197 a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * a(k).
%F A377197 a(n) = (6*n*a(n-1) - 5*(n-2)*a(n-2))/n for n > 1.
%F A377197 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n-1,n-k).
%F A377197 a(n) ~ 16 * sqrt(n) * 5^(n - 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Oct 26 2024
%F A377197 a(n) = 6*hypergeom([5/2, 1-n], [2], -4) for n > 0. - _Stefano Spezia_, May 08 2025
%t A377197 Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, May 11 2025 *)
%o A377197 (PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n-1, n-k));
%o A377197 (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x))^(3/2))); // _Vincenzo Librandi_, May 11 2025
%Y A377197 Cf. A085362, A377199, A377200.
%Y A377197 Cf. A002457, A377198.
%Y A377197 Cf. A374497.
%K A377197 nonn
%O A377197 0,2
%A A377197 _Seiichi Manyama_, Oct 19 2024