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A025759 6th-order Vatalan numbers (generalization of Catalan numbers).

%I A025759 #32 Aug 04 2024 12:43:46
%S A025759 1,1,16,361,9346,260710,7622290,230167345,7116228250,224012186830,
%T A025759 7152402830440,230999414308090,7531444277855740,247510726140787240,
%U A025759 8189274963276187990,272537576338530727585,9116110475685684958810,306286229879232067776310
%N A025759 6th-order Vatalan numbers (generalization of Catalan numbers).
%H A025759 Vincenzo Librandi, <a href="/A025759/b025759.txt">Table of n, a(n) for n = 0..200</a>
%H A025759 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.
%F A025759 G.f.: 6/(5+(1-36*x)^(1/6)).
%F A025759 Recurrence: for n>5,
%F A025759 a(n)=((n-1)*(n-2)*(n-3)*(n-4)*(78119*n-273420)*a(n-1)-90*(n-2)*(n-3)*(n-4)*(62494*n^2-499959*n+1014685)*a(n-2)+180*(n-3)*(n-4)*(1124856*n^3-15185808*n^2+68852647*n-104826890)*a(n-3)-3240*(n-4)*(1124784*n^4-22496184*n^3+169193274*n^2-567111339*n+714764687)*a(n-4)+5184*(5060556*n^5-139170960*n^4+1530231885*n^3-8408803050*n^2+23092951859*n-25356134300)*a(n-5)+93312*(2*n-11)*(3*n-16)*(3*n-17)*(6*n-31)*(6*n-35)*a(n-6))/(434*n*(n-1)*(n-2)*(n-3)*(n-4)). - _Tani Akinari_, Sep 15 2015
%F A025759 a(n) ~ 36^n / (25 * Gamma(5/6) * n^(7/6)) * (1 - 2^(1/3)*sqrt(3)*Gamma(2/3) / (5*sqrt(Pi)*n^(1/6))). - _Vaclav Kotesovec_, Sep 22 2015
%F A025759 a(n) = (-1)^(n+1) * 6^(2*n+1) * Sum_{k>=0} (-1/5)^(k+1) * binomial(k/6,n). - _Seiichi Manyama_, Aug 04 2024
%t A025759 Table[SeriesCoefficient[6/(5 + (1 - 36*x)^(1/6)), {x, 0, n}], {n, 0, 20}] (* _Vincenzo Librandi_, Dec 29 2012 *)
%o A025759 (Maxima) a[0]:1$ a[1]:1$ a[2]:16$ a[3]:361$ a[4]:9346$ a[5]:260710$
%o A025759 a[n]:=((n-1)*(n-2)*(n-3)*(n-4)*(78119*n-273420)*a[n-1]-90*(n-2)*(n-3)*(n-4)*(62494*n^2-499959*n+1014685)*a[n-2]+180*(n-3)*(n-4)*(1124856*n^3-15185808*n^2+68852647*n-104826890)*a[n-3]-3240*(n-4)*(1124784*n^4-22496184*n^3+169193274*n^2-567111339*n+714764687)*a[n-4]+5184*(5060556*n^5-139170960*n^4+1530231885*n^3-8408803050*n^2+23092951859*n-25356134300)*a[n-5]+93312*(2*n-11)*(3*n-16)*(3*n-17)*(6*n-31)*(6*n-35)*a[n-6])/(434*n*(n-1)*(n-2)*(n-3)*(n-4));
%o A025759 makelist(a[n],n,0,500);  /* _Tani Akinari_, Sep 15 2015  */
%o A025759 (PARI) default(seriesprecision, 40); Vec(6/(5+(1-36*x)^(1/6)) + O(x^30)) \\ _Michel Marcus_, Sep 15 2015
%Y A025759 a(n), n >= 1, = row sums of triangle A049224.
%Y A025759 Cf. A000108, A025756, A025757, A025758, A025760, A025761, A025762, A025763.
%K A025759 nonn
%O A025759 0,3
%A A025759 _Olivier Gérard_