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 A127902 #37 Sep 08 2022 08:45:29
%S A127902 1,1,1,1,2,6,16,36,75,163,391,991,2498,6150,15016,37116,93481,238137,
%T A127902 607921,1550401,3959335,10155615,26182267,67753907,175713561,
%U A127902 456422121,1187771521,3097869841,8097629671,21207212047,55628797891,146129168651,384401493333,1012608918421
%N A127902 Series reversion of x/(1 + x + x^4).
%C A127902 Binomial transform of A002293, with three interpolated zeros (series reversion of x/(1+x^4)).
%C A127902 Also the number of rooted labeled trees where each node has 0, 1, or 4 children. - _Patrick Devlin_, Mar 04 2012
%C A127902 Number of lattice paths from (0,0) to (n-1,0) that do not go below the x-axis or above the diagonal x=y and consist of steps u=(1,1), H=(1,0) and D=(1,-3); a(7) = 16: HHHHHH, uuuDHH, HuuuDH, uHuuDH, uuHuDH, uuuHDH, HHuuuD, HuHuuD, uHHuuD, HuuHuD, uHuHuD, uuHHuD, HuuuHD, uHuuHD, uuHuHD, uuuHHD. - _Alois P. Heinz_, Apr 14 2014
%H A127902 Vincenzo Librandi, <a href="/A127902/b127902.txt">Table of n, a(n) for n = 1..200</a>
%H A127902 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%F A127902 a(n) = Sum_{k=0..floor(n/4)} C(n,4k)*C(4k,k)/(3k+1).
%F A127902 Recurrence: 3*(n-1)*(3*n-7)*(3*n+1)*a(n) = 3*(2*n-3)*(18*n^2 - 54*n + 29)*a(n-1) - 3*(n-2)*(54*n^2 - 216*n + 209)*a(n-2) + 54*(n-3)*(n-2)*(2*n-5)*a(n-3) + 229*(n-4)*(n-3)*(n-2)*a(n-4). - _Vaclav Kotesovec_, Aug 20 2013
%F A127902 a(n) ~ sqrt(4+3^(3/4))*3^(1/4) * (1+4/3*3^(1/4))^n /(12*sqrt(Pi/2) *n^(3/2)). - _Vaclav Kotesovec_, Aug 20 2013
%F A127902 G.f. A(x) satisfies: A(x) = x * (1 + A(x) + A(x)^4). - _Ilya Gutkovskiy_, Jul 01 2020
%t A127902 Rest[CoefficientList[InverseSeries[Series[x/(1+x+x^4),{x,0,20}],x],x]] (* _Vaclav Kotesovec_, Aug 20 2013 *)
%o A127902 (PARI) for(n=0, 30, print1(sum(k=0,floor(n/4), binomial(n,4*k) *binomial(4*k,k)/(3*k+1)), ", ")) \\ _G. C. Greubel_, Apr 30 2018
%o A127902 (Magma) [(&+[Binomial(n,4*k)*Binomial(4*k,k)/(3*k+1): k in [0..Floor(n/4)]]): n in [0..30]]; // _G. C. Greubel_, Apr 30 2018
%Y A127902 Cf. A240904.
%K A127902 easy,nonn
%O A127902 1,5
%A A127902 _Paul Barry_, Feb 05 2007
%E A127902 Offset corrected by _Vaclav Kotesovec_, Aug 20 2013
%E A127902 More terms from _Vincenzo Librandi_, Apr 15 2014