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A101602
G.f.: c(3x)^4, c(x) the g.f. of A000108.
2
1, 12, 126, 1296, 13365, 138996, 1459458, 15466464, 165297834, 1780130520, 19301700924, 210564010080, 2309623985565, 25458117777540, 281857732537050, 3133071216411840, 34953325758094590, 391242268149428520, 4392583646950402020, 49454259823789423200
OFFSET
0,2
FORMULA
G.f.: 16/(1+sqrt(1-12x))^4.
a(n)=((8n+12)/(3n+12))((3n+3)/(n+3))3^n*C(n+1).
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
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 1479/484 - 2691*arcsin(1/(2*sqrt(3)))/(121*sqrt(11)).
Sum_{n>=0} (-1)^n/a(n) = 7569*arcsinh(1/(2*sqrt(3)))/(169*sqrt(13)) - 1767/676. (End)
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 08 2004
STATUS
approved