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A081041 6th binomial transform of (1,5,0,0,0,0,0,0,...).

%I A081041 #29 Jan 31 2025 15:06:34
%S A081041 1,11,96,756,5616,40176,279936,1912896,12877056,85660416,564350976,
%T A081041 3688436736,23944605696,154551545856,992612745216,6347497291776,
%U A081041 40435908673536,256721001578496,1624959306694656,10257555623510016
%N A081041 6th binomial transform of (1,5,0,0,0,0,0,0,...).
%H A081041 Vincenzo Librandi, <a href="/A081041/b081041.txt">Table of n, a(n) for n = 0..300</a>
%H A081041 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 A081041 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-36).
%F A081041 a(n) = 12*a(n-1) - 36*a(n-2) for n>1, a(0)=1, a(1)=9.
%F A081041 a(n) = (5*n+6)*6^(n-1).
%F A081041 a(n) = Sum_{k=0..n} (k+1)*5^k*binomial(n, k).
%F A081041 G.f.: (1-x)/(1-6*x)^2.
%F A081041 E.g.f.: exp(6*x)*(1 + 5*x). - _Stefano Spezia_, Jan 31 2025
%t A081041 CoefficientList[Series[(1 - x)/(1 - 6 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t A081041 LinearRecurrence[{12,-36},{1,11},20] (* _Harvey P. Dale_, Mar 04 2019 *)
%o A081041 (Magma) [(5*n+6)*6^(n-1): n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2013
%Y A081041 Cf. A081040, A081042.
%K A081041 nonn,easy
%O A081041 0,2
%A A081041 _Paul Barry_, Mar 04 2003