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A025757 4th-order Vatalan numbers (generalization of Catalan numbers).

%I A025757 #27 Aug 04 2024 12:43:50
%S A025757 1,1,7,69,783,9597,123495,1643397,22413183,311466829,4392857431,
%T A025757 62702224213,903886452975,13138698859677,192337495360071,
%U A025757 2832859169364261,41946319269028191,624009420903043821
%N A025757 4th-order Vatalan numbers (generalization of Catalan numbers).
%H A025757 Vincenzo Librandi, <a href="/A025757/b025757.txt">Table of n, a(n) for n = 0..200</a>
%H A025757 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H A025757 T. M. Richardson, <a href="http://arxiv.org/abs/1410.5880">The Super Patalan Numbers</a>, arXiv preprint arXiv:1410.5880, 2014
%H A025757 T. M. Richardson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Richardson/rich2.html">The Super Patalan Numbers</a>, J. Int. Seq. 18 (2015) # 15.3.3.
%F A025757 G.f.: 4 / (3+(1-16*x)^(1/4)).
%F A025757 a(n) = Sum_{m=1..n-1} (m/n*4^(n-m)) * Sum_{k=1..n-m} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j) + 1. - _Vladimir Kruchinin_, Feb 08 2011
%F A025757 Conjecture: 5*n*(n-1)*(n-2)*a(n) -(239*n-600)*(n-1)*(n-2)*a(n-1) +24*(n-2)*(158*n^2-953*n+1445)*a(n-2) +16*(-1232*n^3+13056*n^2-45949*n+53730)*a(n-3) -128*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - _R. J. Mathar_, Jul 28 2014
%F A025757 a(n) = (-1)^(n+1) * 4^(2*n+1) * Sum_{k>=0} (-1/3)^(k+1) * binomial(k/4,n). - _Seiichi Manyama_, Aug 04 2024
%t A025757 Table[SeriesCoefficient[4/(3 + (1 - 16*x)^(1/4)), {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Dec 29 2012 *)
%Y A025757 a(n), n >= 1, = row sums of triangle A049213.
%Y A025757 Cf. A000108, A025756, A025758, A025759, A025760, A025761, A025762, A025763.
%K A025757 nonn
%O A025757 0,3
%A A025757 _Olivier Gérard_