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A025759
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6th-order Vatalan numbers (generalization of Catalan numbers).
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2
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1, 1, 16, 361, 9346, 260710, 7622290, 230167345, 7116228250, 224012186830, 7152402830440, 230999414308090, 7531444277855740, 247510726140787240, 8189274963276187990, 272537576338530727585, 9116110475685684958810, 306286229879232067776310
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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MATHEMATICA
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Table[SeriesCoefficient[6/(5 + (1 - 36*x)^(1/6)), {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 29 2012 *)
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PROG
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(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));
(PARI) default(seriesprecision, 40); Vec(6/(5+(1-36*x)^(1/6)) + O(x^30)) \\ Michel Marcus, Sep 15 2015
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CROSSREFS
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a(n), n >= 1, = row sums of triangle A049224.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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