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 A101600 #33 Feb 17 2025 22:37:04
%S A101600 1,6,45,378,3402,32076,312741,3127410,31899582,330595668,3471254514,
%T A101600 36848701764,394807518900,4263921204120,46370143094805,
%U A101600 507343918566690,5580783104233590,61682339573108100,684673969261499910,7629224228913856140,85308598196036755020
%N A101600 Expansion of g.f. c(3x)^2, where c(x) is the g.f. of A000108.
%H A101600 Samuele Giraudo, <a href="http://arxiv.org/abs/1603.01394">Pluriassociative algebras II: The polydendriform operad and related operads</a>, arXiv:1603.01394 [math.CO], 2016.
%F A101600 G.f.: 4/(1+sqrt(1-12*x))^2.
%F A101600 a(n) = 3^n * A000108(n+1).
%F A101600 (n+2)*a(n) -6*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 15 2011
%F A101600 O.g.f. A(x) = 1/x*series reversion( x/(1 + 3*x)^2 ). 1 + x*A'(x)/A(x) = 1/sqrt(1 - 12*x) is the o.g.f. for A098658. - _Peter Bala_, Jul 17 2015
%F A101600 From _Amiram Eldar_, May 15 2022: (Start)
%F A101600 Sum_{n>=0} 1/a(n) = 87/121 + 648*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
%F A101600 Sum_{n>=0} (-1)^n/a(n) = 93/169 + 648*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)). (End)
%F A101600 E.g.f.: BesselI(1,6*z)*exp(6*z)/(3*z) where BesselI is the modified Bessel function of type I. - _Karol A. Penson_, Feb 17 2025
%p A101600 Z[0]:=1: for k to 30 do Z[k]:=simplify(1/(1-3*z*Z[k-1])) od: g:=sum((Z[j]-Z[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(coeff(gser, z, n)/3, n=1..19); # _Zerinvary Lajos_, May 21 2008
%t A101600 a[n_] := 3^n * CatalanNumber[n + 1]; Array[a, 20, 0] (* _Amiram Eldar_, May 15 2022 *)
%Y A101600 Cf. A050159, A101601, A101602, A098658.
%K A101600 easy,nonn
%O A101600 0,2
%A A101600 _Paul Barry_, Dec 08 2004