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A081040 5th binomial transform of (1,4,0,0,0,0,...).

%I A081040 #30 Jan 31 2025 15:03:08
%S A081040 1,9,65,425,2625,15625,90625,515625,2890625,16015625,87890625,
%T A081040 478515625,2587890625,13916015625,74462890625,396728515625,
%U A081040 2105712890625,11138916015625,58746337890625,308990478515625,1621246337890625
%N A081040 5th binomial transform of (1,4,0,0,0,0,...).
%H A081040 Vincenzo Librandi, <a href="/A081040/b081040.txt">Table of n, a(n) for n = 0..300</a>
%H A081040 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 A081040 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-25).
%F A081040 a(n) = 10*a(n-1) - 25*a(n-2), a(0)=1, a(1)=9.
%F A081040 a(n) = (4n+5)*5^(n-1).
%F A081040 a(n) = Sum_{k=0..n} (k+1)*4^k*binomial(n, k).
%F A081040 G.f.: (1-x)/(1-5*x)^2.
%F A081040 E.g.f.: exp(5*x)*(1 + 4*x). - _Stefano Spezia_, Jan 31 2025
%t A081040 CoefficientList[Series[(1 - x) / (1 - 5 x)^2, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t A081040 LinearRecurrence[{10,-25},{1,9},30] (* _Harvey P. Dale_, Jan 10 2021 *)
%o A081040 (Magma) [(4*n+5)*5^(n-1): n in [0..25]]; // _Vincenzo Librandi_, Aug 06 2013
%o A081040 (PARI) a(n)=(4*n+5)*5^(n-1) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A081040 Cf. A081039, A081041.
%K A081040 nonn,easy
%O A081040 0,2
%A A081040 _Paul Barry_, Mar 03 2003