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a(n) = C(n)*(7n+1) where C(n)=Catalan numbers (A000108).
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%I #30 Jun 01 2024 11:50:39

%S 1,8,30,110,406,1512,5676,21450,81510,311168,1192516,4585308,17681020,

%T 68346800,264769560,1027653570,3995416710,15557374800,60660114900,

%U 236813267460,925540979220,3621007518960,14179797364200,55575657411300,217993800897756,855702566655552

%N a(n) = C(n)*(7n+1) where C(n)=Catalan numbers (A000108).

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H Andrew Howroyd, <a href="/A050477/b050477.txt">Table of n, a(n) for n = 0..200</a>

%F 3*(n+1)*a(n) + (-17*n-1)*a(n-1) + 10*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Feb 13 2015

%F -(n+1)*(7*n-6)*a(n) + 2*(7*n+1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Feb 13 2015

%F G.f.: (3 - 5*x - 3*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - _Ilya Gutkovskiy_, Jun 13 2017

%t Table[CatalanNumber[n](7n+1),{n,0,30}] (* _Harvey P. Dale_, Jun 01 2024 *)

%o (PARI) a(n) = (7*n+1) * binomial(2*n,n)/(n+1) \\ _Michel Marcus_, Jul 24 2013

%o (Magma) [Catalan(n)*(7*n+1):n in [0..25] ]; // _Marius A. Burtea_, Jan 05 2020

%o (Magma) R<x>:=PowerSeriesRing(Rationals(),27); (Coefficients(R!( (3-5*x-3*Sqrt(1-4*x))/(x*Sqrt(1 - 4*x))) )); // _Marius A. Burtea_, Jan 05 2020

%Y Column k=7 of A330965.

%Y Cf. A016921, A000108, A051945.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Dec 24 1999

%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 02 2020