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Sum of values of vertices of type A at level n of the hyperbolic Pascal pyramid.
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%I #15 Sep 18 2017 08:03:53

%S 0,0,6,18,54,174,582,1974,6726,22950,78342,267462,913158,3117702,

%T 10644486,36342534,124081158,423639558,1446395910,4938304518,

%U 16860426246,57565095942,196539531270,671027933190,2291032670214,7822074814470,26706233917446,91180786040838

%N Sum of values of vertices of type A at level n of the hyperbolic Pascal pyramid.

%H Colin Barker, <a href="/A292295/b292295.txt">Table of n, a(n) for n = 0..1000</a>

%H László Németh, <a href="http://arxiv.org/abs/1511.02067">Hyperbolic Pascal pyramid</a>, arXiv:1511.0267 [math.CO], 2015 (1st line of Table 2).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,2).

%F a(n) = 5*a(n-1) - 6*a(n-2) + 2*a(n-3), n >= 4.

%F From _Colin Barker_, Sep 17 2017: (Start)

%F G.f.: 6*x^2*(1 - 2*x) / ((1 - x)*(1 - 4*x + 2*x^2)).

%F a(n) = (-3/2)*(-4 + (4-3*sqrt(2))*(2+sqrt(2))^n + (2-sqrt(2))^n*(4+3*sqrt(2))) for n>0.

%F (End)

%t CoefficientList[Series[6*x^2*(1 - 2*x)/((1 - x)*(1 - 4*x + 2*x^2)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Sep 17 2017 *)

%o (PARI) concat(vector(2), Vec(6*x^2*(1 - 2*x) / ((1 - x)*(1 - 4*x + 2*x^2)) + O(x^30))) \\ _Colin Barker_, Sep 17 2017

%Y Cf. A264237.

%K nonn,easy

%O 0,3

%A _Eric M. Schmidt_, Sep 13 2017