login
4th binomial transform of (1,3,0,0,0,0,0,.....).
5

%I #22 Sep 08 2022 08:45:09

%S 1,7,40,208,1024,4864,22528,102400,458752,2031616,8912896,38797312,

%T 167772160,721420288,3087007744,13153337344,55834574848,236223201280,

%U 996432412672,4191888080896,17592186044416,73667279060992

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

%H Vincenzo Librandi, <a href="/A081039/b081039.txt">Table of n, a(n) for n = 0..300</a>

%H Silvana Ramaj, <a href="https://digitalcommons.georgiasouthern.edu/cgi/viewcontent.cgi?article=3464&amp;context=etd">New Results on Cyclic Compositions and Multicompositions</a>, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).

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

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

%F a(n) = sum( k=0..n, (k+1)*3^k*binomial(n, k) ).

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

%t CoefficientList[Series[(1 - x)/(1 - 4 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)

%t LinearRecurrence[{8,-16},{1,7},30] (* _Harvey P. Dale_, Dec 13 2015 *)

%o (Magma) [(3*n+4)*4^(n-1): n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2013

%o (PARI) a(n)=(3*n+4)*4^(n-1) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A081038, A081040.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 03 2003