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A129442 Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.

%I A129442 #45 Feb 06 2024 08:13:46
%S A129442 1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,11485068,
%T A129442 54377288,259663576,1249249981,6049846848,29469261934,144293491564,
%U A129442 709806846980,3506278661820,17385618278700,86500622296800
%N A129442 Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.
%C A129442 The sequence b(n) = [0,1,2,6,21,80,322,1348,...] for n >= 0 is the Catalan transform of Catalan numbers C(n-1), with C(-1)=0; Sum_{k=0..n} A106566(n,k) * A000108(k-1) = b(n).
%C A129442 A121988 is an essentially identical sequence. - _R. J. Mathar_, Jun 13 2008
%C A129442 Catalan transform of A014137. - _R. J. Mathar_, Nov 11 2008
%H A129442 Vincenzo Librandi, <a href="/A129442/b129442.txt">Table of n, a(n) for n = 0..300</a>
%H A129442 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%F A129442 a(n-1) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1, n-1)*binomial(2*k-2, k-1).
%F A129442 G.f.: (1-sqrt(2*sqrt(1-4*x)-1))/(2*x). - _Emeric Deutsch_, Jun 20 2007 Corrected by Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
%F A129442 From _Vaclav Kotesovec_, Oct 20 2012: (Start)
%F A129442 Recurrence: 3*n*(n+1)*a(n) = 14*n*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
%F A129442 a(n) ~ 2^(4*n+3/2)/(3^(n+1/2)*sqrt(Pi)*n^(3/2)). (End)
%F A129442 0 = +a(n)*(+a(n+1)*(+262144*a(n+2) -275968*a(n+3) +52608*a(n+4)) +a(n+2)*(-50176*a(n+2) +107680*a(n+3) -27930*a(n+4)) +a(n+3)*(-6006*a(n+3) +2574*a(n+4))) +a(n+1)*(+a(n+1)*(-17920*a(n+2) +21952*a(n+3) -4494*a(n+4)) +a(n+2)*(+5152*a(n+2) -15820*a(n+3) +4611*a(n+4)) +a(n+3)*(+1470*a(n+3) -630*a(n+4))) +a(n+2)*(+a(n+2)*(+42*a(n+2) +129*a(n+3) -63*a(n+4)) +a(n+3)*(-63*a(n+3) +27*a(n+4))) for n>=0. - _Michael Somos_, May 28 2023
%F A129442 From _Seiichi Manyama_, Jan 10 2023: (Start)
%F A129442 G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2) ).
%F A129442 a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-2*k). (End)
%e A129442 G.f. = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 322*x^5 + 1349*x^6 + ... - _Michael Somos_, May 28 2023
%p A129442 c := proc (x) options operator, arrow; (1/2)*(1-sqrt(1-4*x))/x end proc; G := simplify(c(x)*c(x*c(x))); Gser := series(G, x = 0, 28); seq(coeff(Gser, x, n), n = 0 .. 24) # _Emeric Deutsch_, Jun 20 2007
%t A129442 a[n_]:= Sum[ Binomial[2n -k-1, n-1]*Binomial[2k-2, k-1], {k, n}]/n;
%t A129442 Array[a, 23] (* _Robert G. Wilson v_, Jul 18 2007 *)
%o A129442 (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // _G. C. Greubel_, Feb 06 2024
%o A129442 (SageMath)
%o A129442 def A129442_list(prec):
%o A129442     P.<x> = PowerSeriesRing(ZZ, prec)
%o A129442     return P( (1-sqrt(2*sqrt(1-4*x)-1))/(2*x) ).list()
%o A129442 A129442_list(40) # _G. C. Greubel_, Feb 06 2024
%Y A129442 Cf. A000108, A014137, A106566, A121988, A236339.
%K A129442 nonn
%O A129442 0,2
%A A129442 _Philippe Deléham_, May 28 2007, Jun 20 2007
%E A129442 More terms from _Emeric Deutsch_, Jun 20 2007