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

0,5

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A264236.

Sequence in context: A212664 A342969 A050392 * A366995 A191468 A203243

Adjacent sequences: A292291 A292292 A292293 * A292295 A292296 A292297

Keyword

nonn,easy

Author

Eric M. Schmidt, Sep 13 2017

Status

approved