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

1, 1, 16, 361, 9346, 260710, 7622290, 230167345, 7116228250, 224012186830, 7152402830440, 230999414308090, 7531444277855740, 247510726140787240, 8189274963276187990, 272537576338530727585, 9116110475685684958810, 306286229879232067776310

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

Author

Olivier Gérard

Status

approved