A292294 Number of vertices of type E at level n of the hyperbolic Pascal pyramid.

0, 0, 0, 0, 3, 39, 357, 2952, 23622, 186984, 1474773, 11617815, 91485075, 720308160, 5671099008, 44648794944, 351520074867, 2767513935927, 21788596994037, 171541276628904, 1350541654293318, 10632792057873480, 83711795070905925, 659061569195852295

Offset: 0

Author

Eric M. Schmidt, Sep 13 2017

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• László Németh, Hyperbolic Pascal pyramid, arXiv:1511.0267 [math.CO], 2015 (5th line of Table 1).
• Index entries for linear recurrences with constant coefficients, signature (12,-37,37,-12,1).

Crossrefs

Cf. A264236.

Programs

Mathematica

CoefficientList[Series[3*x^4*(1 + x)/((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2017 *)
LinearRecurrence[{12,-37,37,-12,1},{0,0,0,0,3,39},30] (* Harvey P. Dale, Oct 09 2018 *)

PARI

concat(vector(4), Vec(3*x^4*(1 + x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Sep 17 2017

Formula

a(n) = 12*a(n-1) - 37*a(n-2) + 37*a(n-3) - 12*a(n-4) + a(n-5), n >= 6.

G.f.: 3*x^4*(1 + x) / ((1 - x)*(1 - 8*x + x^2)*(1 - 3*x + x^2)). - Colin Barker, Sep 17 2017

a(n) = 1 + (A001091(n-2) - 3*Lucas(2*(2-n)))/5 for n > 0. - Ehren Metcalfe, Apr 18 2019