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

%I A081039 #26 Jan 31 2025 14:56:15
%S A081039 1,7,40,208,1024,4864,22528,102400,458752,2031616,8912896,38797312,
%T A081039 167772160,721420288,3087007744,13153337344,55834574848,236223201280,
%U A081039 996432412672,4191888080896,17592186044416,73667279060992
%N A081039 4th binomial transform of (1,3,0,0,0,0,0,.....).
%H A081039 Vincenzo Librandi, <a href="/A081039/b081039.txt">Table of n, a(n) for n = 0..300</a>
%H A081039 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 A081039 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).
%F A081039 a(n) = 8*a(n-1) -16*a(n-2) with n>1, a(0)=1, a(1)=7.
%F A081039 a(n) = (3*n+4)*4^(n-1).
%F A081039 a(n) = Sum_{k=0..n} (k+1)*3^k*binomial(n, k).
%F A081039 G.f.: (1-x)/(1-4*x)^2.
%F A081039 E.g.f.: exp(4*x)*(1 + 3*x). - _Stefano Spezia_, Jan 31 2025
%t A081039 CoefficientList[Series[(1 - x)/(1 - 4 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t A081039 LinearRecurrence[{8,-16},{1,7},30] (* _Harvey P. Dale_, Dec 13 2015 *)
%o A081039 (Magma) [(3*n+4)*4^(n-1): n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2013
%o A081039 (PARI) a(n)=(3*n+4)*4^(n-1) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A081039 Cf. A081038, A081040.
%K A081039 nonn,easy
%O A081039 0,2
%A A081039 _Paul Barry_, Mar 03 2003