A101602 - OEIS

%I #15 May 15 2022 07:31:31

%S 1,12,126,1296,13365,138996,1459458,15466464,165297834,1780130520,

%T 19301700924,210564010080,2309623985565,25458117777540,

%U 281857732537050,3133071216411840,34953325758094590,391242268149428520,4392583646950402020,49454259823789423200

%N G.f.: c(3x)^4, c(x) the g.f. of A000108.

%F G.f.: 16/(1+sqrt(1-12x))^4.

%F a(n)=((8n+12)/(3n+12))((3n+3)/(n+3))3^n*C(n+1).

%F Conjecture: (n+4)*a(n) - 6*(3*n+7)*a(n-1) + 36*(2*n+1)*a(n-2) = 0. - _R. J. Mathar_, Nov 15 2011

%F From _Amiram Eldar_, May 15 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 1479/484 - 2691*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).

%F Sum_{n>=0} (-1)^n/a(n) = 7569*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)) - 1767/676. (End)

%t a[n_] := ((8*n + 12)/(3*n + 12)) * ((3*n + 3)/(n + 3))* 3^n* CatalanNumber[n + 1]; Array[a, 20, 0] (* _Amiram Eldar_, May 15 2022 *)

%Y Cf. A000108, A050159, A101600, A101601.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Dec 08 2004