A081915 a(n) = 4^n*(n^3 - 3*n^2 + 2*n + 384)/384.

1, 4, 16, 65, 272, 1184, 5376, 25344, 122880, 606208, 3014656, 15007744, 74448896, 367001600, 1795162112, 8707375104, 41875931136, 199715979264, 944892805120, 4436701216768, 20684562497536, 95794950569984, 440904162738176, 2017603836968960, 9183121115185152

Offset

0,2

Comments

Binomial transform of A081914. 4th binomial transform of (1,0,0,1,0,0,0,0,...). Case k=4 where a(n,k) = k^n*(n^3 - 3*n^2 + 2*n + 6*k^3)/(6*k^3), with g.f. (1 - 3*k*x + 3*k^2*x^2 - (k^3-1)*x^3)/(1-k*x)^4.

Links

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

Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256)

Formula

a(n) = 4^n*(n^3 - 3*n^2 + 2*n + 384)/384.

G.f.: (1 - 12*x + 48*x^2 - 63*x^3)/(1-4*x)^4.

From Elmo R. Oliveira, Nov 12 2025: (Start)

E.g.f.: (1 + x^3/6)*exp(4*x).

a(n) = 16*a(n-1) - 96*a(n-2) + 256*a(n-3) - 256*a(n-4). (End)

Mathematica

LinearRecurrence[{16, -96, 256, -256}, {1, 4, 16, 65}, 30] (* Harvey P. Dale, Aug 14 2017 *)
CoefficientList[Series[(1 - 12x + 48x^2 - 63x^3)/(1-4x)^4 , {x, 0, 30}], x] (* Stefano Spezia, Sep 02 2018 *)

Prog

(Magma) [4^n*(n^3-3*n^2+2*n+384)/384: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
(PARI) a(n)=4^n*(n^3-3*n^2+2*n+384)/384 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A081914, A081916.

Sequence in context: A026674 A099781 A026872 * A307878 A217632 A026762

Adjacent sequences: A081912 A081913 A081914 * A081916 A081917 A081918

Keyword

easy,nonn

Author

Paul Barry, Mar 31 2003

Status

approved