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 A382542 #17 May 13 2025 02:42:26
%S A382542 1,3,24,172,1191,8091,54214,359274,2358945,15365815,99399132,
%T A382542 639081780,4086689187,26006041209,164767882902,1039787209898,
%U A382542 6537976304109,40973438195025,255998969164612,1594973077037136,9911483124031335,61443351455986359,380044418794190118
%N A382542 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^3.
%H A382542 Vincenzo Librandi, <a href="/A382542/b382542.txt">Table of n, a(n) for n = 0..600</a>
%F A382542 a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n+k/2-1,n-k).
%F A382542 D-finite with recurrence (1951*n-648)*(n-1)*a(n) +2*(-19507*n^2+55239*n-42212)*a(n-1) +(310113*n^2-1357025*n+1557754)*a(n-2) +2*(-616231*n^2+3607803*n-5547102)*a(n-3) +4*(616138*n^2-4491203*n+8306122)*a(n-4) -256*(3899*n-16903)*(2*n-9)*a(n-5)=0. - _R. J. Mathar_, Apr 02 2025
%t A382542 Table[Sum[(4)^(n-k)* Binomial[k+2,2]*Binomial[n+k/2-1, n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, May 12 2025 *)
%o A382542 (PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(n+k/2-1, n-k));
%o A382542 (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(3/2))^3)); // _Vincenzo Librandi_, May 12 2025
%Y A382542 Cf. A382514, A382541.
%K A382542 nonn,easy
%O A382542 0,2
%A A382542 _Seiichi Manyama_, Mar 31 2025