A081914 - OEIS

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A081914

a(n) = 3^n*(n^3 - 3n^2 + 2n + 162)/162.

1, 3, 9, 28, 93, 333, 1269, 5022, 20169, 80919, 321489, 1259712, 4861701, 18482337, 69264477, 256154562, 935867601, 3381559083, 12096128601, 42874534116, 150706570221, 525729603573, 1821263718789, 6269238352998, 21454184419353

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OFFSET

0,2

COMMENTS

Binomial transform of A081913 3rd binomial transform of (1,0,0,1,0,0,0,0,...). Case k=3 where a(n,k) = k^n*(n^3 - 3n^2 + 2n + 6k^3)/(6k^3), with g.f. (1 - 3kx + 3k^2x^2 - (k^3-1)x^3)/(1-kx)^4.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..150

Index entries for linear recurrences with constant coefficients, signature (12,-54,108,-81).

FORMULA

a(n) = 3^n*(n^3 - 3n^2 + 2n + 162)/162.

G.f.: (1 - 9x + 27x^2 - 26x^3)/(1-3x)^4.

MATHEMATICA

CoefficientList[Series[(1 - 9x + 27x^2 - 26x^3)/(1-3x)^4 , {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

LinearRecurrence[{12, -54, 108, -81}, {1, 3, 9, 28}, 30] (* Harvey P. Dale, Aug 01 2019 *)

PROG

(Magma) [3^n*(n^3-3*n^2+2*n+162)/162: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

(PARI) a(n)=3^n*(n^3-3*n^2+2*n+162)/162 \\ Charles R Greathouse IV, Oct 07 2015