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 A025759 6th-order Vatalan numbers (generalization of Catalan numbers). 9
 1, 1, 16, 361, 9346, 260710, 7622290, 230167345, 7116228250, 224012186830, 7152402830440, 230999414308090, 7531444277855740, 247510726140787240, 8189274963276187990, 272537576338530727585, 9116110475685684958810, 306286229879232067776310 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. FORMULA G.f.: 6/(5+(1-36*x)^(1/6)). Recurrence: for n>5, 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 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 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 MATHEMATICA Table[SeriesCoefficient[6/(5 + (1 - 36*x)^(1/6)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *) PROG (Maxima) a[0]:1\$ a[1]:1\$ a[2]:16\$ a[3]:361\$ a[4]:9346\$ a[5]:260710\$ 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)); makelist(a[n], n, 0, 500); /* Tani Akinari, Sep 15 2015 */ (PARI) default(seriesprecision, 40); Vec(6/(5+(1-36*x)^(1/6)) + O(x^30)) \\ Michel Marcus, Sep 15 2015 CROSSREFS a(n), n >= 1, = row sums of triangle A049224. Cf. A000108, A025756, A025757, A025758, A025760, A025761, A025762, A025763. Sequence in context: A094101 A034673 A000488 * A276257 A276097 A265476 Adjacent sequences: A025756 A025757 A025758 * A025760 A025761 A025762 KEYWORD nonn,changed AUTHOR Olivier Gérard STATUS approved

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Last modified August 8 18:48 EDT 2024. Contains 375023 sequences. (Running on oeis4.)