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A081187 5th binomial transform of (1,0,1,0,1,...), A059841.

%I A081187 #42 Jan 13 2024 20:32:29
%S A081187 1,5,26,140,776,4400,25376,148160,872576,5169920,30757376,183495680,
%T A081187 1096779776,6563901440,39316299776,235629363200,1412702437376,
%U A081187 8471919656960,50814338072576,304817308958720,1828628975845376
%N A081187 5th binomial transform of (1,0,1,0,1,...), A059841.
%C A081187 Binomial transform of A081187.
%C A081187 a(n) is also the number of words of length n over an alphabet of six letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs, for the 1- to 11-letter word cases. - _Wolfdieter Lang_, Jul 17 2017
%H A081187 Vincenzo Librandi, <a href="/A081187/b081187.txt">Table of n, a(n) for n = 0..200</a>
%H A081187 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-24).
%F A081187 a(n) = 10*a(n-1) - 24*a(n-2) with n > 1, a(0)=1, a(1)=5.
%F A081187 G.f.: (1-5*x)/((1-4*x)*(1-6*x)).
%F A081187 E.g.f.: exp(5*x)*cosh(x).
%F A081187 a(n) = (4^n + 6^n)/2.
%F A081187 a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*5^(n-2k).
%F A081187 E.g.f.: exp(5*x)*cosh(x) = (1/2)*E(0), where E(k) = 1 + 2^k/(3^k - 6*x*9^k/(6*x*3^k + (k+1)*2^k/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2011
%F A081187 a(n) = A074612(n)/2. - _G. C. Greubel_, Jan 13 2024
%p A081187 seq( (4^n + 6^n)/2, n=0..25); # _G. C. Greubel_, Dec 26 2019
%t A081187 CoefficientList[Series[(1-5x)/((1-4x)(1-6x)), {x, 0, 25}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)
%t A081187 LinearRecurrence[{10,-24}, {1,5}, 26] (* _G. C. Greubel_, Dec 26 2019 *)
%o A081187 (Magma) [4^n/2 + 6^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013
%o A081187 (PARI) vector(26, n, (4^(n-1) + 6^(n-1))/2) \\ _G. C. Greubel_, Dec 26 2019
%o A081187 (Sage) [(4^n + 6^n)/2 for n in (0..25)] # _G. C. Greubel_, Dec 26 2019
%o A081187 (GAP) List([0..25], n-> (4^n + 6^n)/2); # _G. C. Greubel_, Dec 26 2019
%Y A081187 Cf. A007582, A074612, A081189.
%K A081187 nonn,easy
%O A081187 0,2
%A A081187 _Paul Barry_, Mar 11 2003