A081039 4th binomial transform of (1,3,0,0,0,0,0,.....).

1, 7, 40, 208, 1024, 4864, 22528, 102400, 458752, 2031616, 8912896, 38797312, 167772160, 721420288, 3087007744, 13153337344, 55834574848, 236223201280, 996432412672, 4191888080896, 17592186044416, 73667279060992

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.

Index entries for linear recurrences with constant coefficients, signature (8,-16).

Formula

a(n) = 8*a(n-1) -16*a(n-2) with n>1, a(0)=1, a(1)=7.

a(n) = (3*n+4)*4^(n-1).

a(n) = Sum_{k=0..n} (k+1)*3^k*binomial(n, k).

G.f.: (1-x)/(1-4*x)^2.

E.g.f.: exp(4*x)*(1 + 3*x). - Stefano Spezia, Jan 31 2025

Mathematica

CoefficientList[Series[(1 - x)/(1 - 4 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{8, -16}, {1, 7}, 30] (* Harvey P. Dale, Dec 13 2015 *)

Prog

(Magma) [(3*n+4)*4^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
(PARI) a(n)=(3*n+4)*4^(n-1) \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A081038, A081040.

Sequence in context: A093737 A268402 A351529 * A227748 A349594 A083327

Adjacent sequences: A081036 A081037 A081038 * A081040 A081041 A081042

Keyword

nonn,easy,changed

Author

Paul Barry, Mar 03 2003

Status

approved