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A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.

%I A007854 #77 Jun 01 2025 04:46:28
%S A007854 1,3,12,51,222,978,4338,19323,86310,386250,1730832,7763550,34847796,
%T A007854 156503064,703149438,3160160811,14206181382,63874779714,287242041528,
%U A007854 1291872728826,5810776384932,26138647551564,117587214581508
%N A007854 Expansion of 1/(1 - 3*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.
%C A007854 Chains in rooted plane trees on n nodes.
%C A007854 The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - _Paul Barry_, Jan 20 2007
%C A007854 Binomial transform of A112657. - _Philippe Deléham_, Nov 25 2007
%C A007854 Row sums of the Riordan matrix (1/sqrt(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))) (A116395). - _Emanuele Munarini_, Apr 26 2011
%C A007854 Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from C(x) = 1/(1 - x*C(x)) = 1/(1 - 3*x*C(x)) (mod 2). - _Peter Bala_, Jul 24 2016
%H A007854 Vincenzo Librandi, <a href="/A007854/b007854.txt">Table of n, a(n) for n = 0..1000</a>
%H A007854 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A007854 Martin Klazar, <a href="http://dx.doi.org/10.1006/eujc.1995.0095">Twelve countings with rooted plane trees</a>, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%H A007854 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0405597">Free quasi-symmetric functions of arbitrary level</a>, arXiv:math/0405597 [math.CO], 2004.
%H A007854 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A007854 a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1) + A067336(n-1) = A067347(3, n-1). - _Henry Bottomley_, Jan 16 2002
%F A007854 a(n) = Sum_{k>=0} A106566(n, k)*3^k. - _Philippe Deléham_, Aug 11 2005
%F A007854 The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - _Philippe Deléham_, Nov 26 2006
%F A007854 a(n) = Sum_{k = 0..n} C(2n,n-k)(2k+1)2^k/(n+k+1). - _Paul Barry_, Jan 20 2007
%F A007854 a(n) = Sum_{k = 0..n} A039599(n,k)*2^k. - _Philippe Deléham_, Sep 08 2007
%F A007854 a(n) = Sum_{k = 0..n} A116395(n,k). - _Vladimir Kruchinin_, Mar 09 2011
%F A007854 From _Emanuele Munarini_, Apr 26 2011 (Start)
%F A007854 a(n) = Sum_{k = 1..n} C(2*n-k,n-k)*(k*3^k)/(2*n-k), for n>0.
%F A007854 a(n) = (1/4)*(9/2)^n-3*Sum_{k=0..n} C(2*k,k)/(2k-1)*(9/2)^(n-k).
%F A007854 D-finite with recurrence: 2*(n+2)*a(n+2)-(17*n+22)*a(n+1)+18*(2*n+1)*a(n)=0. (End)
%F A007854 From _Gary W. Adamson_, Jul 14 2011: (Start)
%F A007854 a(n) = upper left term in M^n, M = the infinite square production matrix:
%F A007854   3, 3, 0, 0, 0, 0, ...
%F A007854   1, 1, 1, 0, 0, 0, ...
%F A007854   1, 1, 1, 1, 0, 0, ...
%F A007854   1, 1, 1, 1, 1, 0, ...
%F A007854   1, 1, 1, 1, 1, 1, ...
%F A007854   ... (End)
%t A007854 CoefficientList[Series[(1+3Sqrt[1-4x])/(4-18x),{x,0,25}],x] (* _Emanuele Munarini_, Apr 26 2011 *)
%t A007854 nm = 25; t = NestList[Append[Accumulate[#], 3 Total[#]] &, {1}, nm];
%t A007854 Table[t[[n, n]], {n, nm}] (*similar to generating Catalan's triangle A009766*)
%t A007854 (* _Li Han_, Oct 23 2020 *)
%o A007854 (Maxima) makelist(kron_delta(n,0)+sum(binomial(2*n-k,n-k)*(k*3^k)/(2*n-k),k,1,n),n,0,12); /* _Emanuele Munarini_, Apr 26 2011 */
%Y A007854 Cf. A000108, A000984, A076035, A076036, A067347, A116395, A126694.
%K A007854 nonn,easy
%O A007854 0,2
%A A007854 _Martin Klazar_
%E A007854 More terms from _Henry Bottomley_, Jan 16 2002