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 A377198 #25 May 11 2025 01:12:21
%S A377198 1,6,42,278,1794,11382,71338,443046,2732034,16751462,102235050,
%T A377198 621535158,3766261506,22758222294,137186860842,825211984710,
%U A377198 4954574749698,29697908825286,177746214414634,1062416305340502,6342559258130178,37823152988963126,225328426205608362
%N A377198 Expansion of 1/(1 - 4*x/(1-x)^2)^(3/2).
%H A377198 Stefano Spezia, <a href="/A377198/b377198.txt">Table of n, a(n) for n = 0..1200</a>
%F A377198 a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * (n-k) * a(k).
%F A377198 a(n) = ((7*n-1)*a(n-1) - (7*n-20)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
%F A377198 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+k-1,n-k).
%F A377198 a(n) ~ 2^(1/4) * sqrt(n) * (1 + sqrt(2))^(2*n) / sqrt(Pi). - _Vaclav Kotesovec_, May 03 2025
%F A377198 a(n) = 6*n*hypergeom([5/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - _Stefano Spezia_, May 08 2025
%t A377198 Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n+k-1,n-k],{k,0,n}],{n,0,35}] (* _Vincenzo Librandi_, May 11 2025 *)
%o A377198 (PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+k-1, n-k));
%o A377198 (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(3/2))); // _Vincenzo Librandi_, May 11 2025
%Y A377198 Cf. A002457, A377197.
%Y A377198 Cf. A377194.
%K A377198 nonn
%O A377198 0,2
%A A377198 _Seiichi Manyama_, Oct 19 2024