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Sum of values of vertices of type B at level n of the hyperbolic Pascal pyramid.
1

%I #16 Sep 18 2017 08:04:00

%S 0,0,0,6,30,114,402,1386,4746,16218,55386,189114,645690,2204538,

%T 7526778,25698042,87738618,299558394,1022756346,3491908602,

%U 11922121722,40704669690,138974435322,474488401914,1620004737018,5531042144250,18884159102970,64474552123386

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

%H Colin Barker, <a href="/A292296/b292296.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 (2nd 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^3 / ((1 - x)*(1 - 4*x + 2*x^2)).

%F a(n) = (1/2)*(-12 + (9-6*sqrt(2))*(2+sqrt(2))^n + (2-sqrt(2))^n*(9+6*sqrt(2))) for n>0.

%F (End)

%t CoefficientList[Series[6*x^3/((1 - x)*(1 - 4*x + 2*x^2)), {x, 0, 30}],

%t x] (* _Wesley Ivan Hurt_, Sep 17 2017 *)

%o (PARI) concat(vector(3), Vec(6*x^3 / ((1 - x)*(1 - 4*x + 2*x^2)) + O(x^30))) \\ _Colin Barker_, Sep 17 2017

%Y Cf. A264237.

%K nonn,easy

%O 0,4

%A _Eric M. Schmidt_, Sep 14 2017