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A081199 5th binomial transform of (0,1,0,1,...), A000035.

%I A081199 #38 Aug 27 2025 07:27:54
%S A081199 0,1,10,76,520,3376,21280,131776,807040,4907776,29708800,179301376,
%T A081199 1080002560,6496792576,39047864320,234555621376,1408407470080,
%U A081199 8454739787776,50745618595840,304542431051776,1827529464217600,10966276296933376,65802055828111360,394829927154712576
%N A081199 5th binomial transform of (0,1,0,1,...), A000035.
%C A081199 Binomial transform of A005059.
%C A081199 Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - _R. H. Hardin_, May 24 2010
%C A081199 a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - _Wolfdieter Lang_, Jul 16 2017
%H A081199 Vincenzo Librandi, <a href="/A081199/b081199.txt">Table of n, a(n) for n = 0..200</a>
%H A081199 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-24).
%F A081199 a(n) = 10*a(n-1) - 24*a(n-2) with n>1, a(0)=0, a(1)=1.
%F A081199 G.f.: x/((1-4*x)*(1-6*x)).
%F A081199 a(n) = 6^n/2 - 4^n/2.
%F A081199 E.g.f.: exp(4*x)*(exp(2*x) - 1)/2. - _Stefano Spezia_, Jul 23 2024
%p A081199 seq(add(2^(2*n-k)*binomial(n,k)/2,k=1..n),n=0..20); # _Zerinvary Lajos_, Apr 18 2009
%t A081199 CoefficientList[Series[x / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)
%t A081199 LinearRecurrence[{10, -24}, {0, 1}, 21] (* _Michael De Vlieger_, Jul 16 2017 *)
%o A081199 (Magma) [6^n/2-4^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013
%Y A081199 Cf. A000035, A003462, A005059, A006516, A081200 (binomial transform of a(n), and 7-letter case), A131577.
%Y A081199 Apart from offset the same as A016149.
%K A081199 nonn,easy,changed
%O A081199 0,3
%A A081199 _Paul Barry_, Mar 11 2003