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A025756 3rd-order Vatalan numbers (generalization of Catalan numbers).

%I A025756 #41 Aug 04 2024 12:43:54
%S A025756 1,1,4,22,139,949,6808,50548,384916,2988418,23559826,188061592,
%T A025756 1516680130,12337999870,101111413540,833914857316,6916004156083,
%U A025756 57638242134229,482444724374734,4053815358183454,34181335453533439
%N A025756 3rd-order Vatalan numbers (generalization of Catalan numbers).
%H A025756 Vincenzo Librandi, <a href="/A025756/b025756.txt">Table of n, a(n) for n = 0..200</a>
%H A025756 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A025756 T. M. Richardson, <a href="http://arxiv.org/abs/1410.5880">The Super Patalan Numbers</a>, arXiv preprint arXiv:1410.5880, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">J. Int. Seq. 18 (2015) # 15.3.3</a>
%F A025756 G.f.: 3 / (2+(1-9*x)^(1/3)).
%F A025756 a(n) = Sum_{m=1..n-1} (m/n) * Sum_{k=1..n-m} binomial(k,n-m-k) * 3^k * (-1)^(n-m-k) * binomial(n+k-1,n-1) + 1. - _Vladimir Kruchinin_, Feb 08 2011
%F A025756 Conjecture: n*(n-1)*a(n) -(n-1)*(19*n-36)*a(n-1) +9*(11*n^2-51*n+60)*a(n-2) -9*(3*n-7)*(3*n-8)*a(n-3) = 0. - _R. J. Mathar_, Nov 14 2011
%F A025756 a(n) ~ 9^n/(4*Gamma(2/3)*n^(4/3)). - _Vaclav Kotesovec_, Oct 08 2012
%F A025756 a(n) = (-1)^(n+1) * 3^(2*n+1) * Sum_{k>=0} (-1/2)^(k+1) * binomial(k/3,n). - _Seiichi Manyama_, Aug 04 2024
%p A025756 A025756 := proc(n)
%p A025756     coeftayl( 3/(2+(1-9*x)^(1/3)), x=0, n);
%p A025756 end proc:
%p A025756 seq(A025756(n), n=0..30); # _Wesley Ivan Hurt_, Aug 02 2014
%t A025756 Table[SeriesCoefficient[3/(2+(1-9*x)^(1/3)),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%o A025756 (Maxima) a[0]:1$ a[n]:=(1/n)*((9*n-6)*a[n-1]-2*sum(a[k]*a[n-1-k], k, 0, n-1))$ makelist(a[n],n,0,1000); /* _Tani Akinari_, Aug 02 2014 */
%Y A025756 Row sums of triangle A048966, n > 0.
%Y A025756 Cf. A000108, A025757, A025758, A025759, A025760, A025761, A025762, A025763.
%K A025756 nonn
%O A025756 0,3
%A A025756 _Olivier Gérard_