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%I #52 Jul 07 2022 23:29:32
%S 1,3,18,108,648,3888,23328,139968,839808,5038848,30233088,181398528,
%T 1088391168,6530347008,39182082048,235092492288,1410554953728,
%U 8463329722368,50779978334208,304679870005248,1828079220031488,10968475320188928,65810851921133568
%N Expansion of exp(3*x)*cosh(3*x).
%C Binomial transform of A081340. 3rd binomial transform of (1,0,9,0,81,0,729,0,...).
%C For m > 1, n > 0, A166469(A002110(m)*a(n)) = (n+1)*A000045(m+1). For n > 0, A166469(a(n)) = 2n. - _Matthew Vandermast_, Nov 05 2009
%C Number of compositions of even natural numbers in n parts <= 5. - _Adi Dani_, May 29 2011
%H Vincenzo Librandi, <a href="/A081341/b081341.txt">Table of n, a(n) for n = 0..125</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (6).
%F a(0)=1, a(n) = 6^n/2, n > 0.
%F G.f.: (1-3*x)/(1-6*x).
%F E.g.f.: exp(3*x)*cosh(3*x).
%F a(n) = A000244(n)*A011782(n). - _Philippe Deléham_, Dec 01 2008
%F a(n) = ((3+sqrt(9))^n + (3-sqrt(9))^n/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
%F a(n) = Sum_{k=0..n} A134309(n,k)*3^k = Sum_{k=0..n} A055372(n,k)*2^k. - _Philippe Deléham_, Feb 04 2012
%F From _Sergei N. Gladkovskii_, Jul 19 2012: (Start)
%F a(n) = ((8*n-4)*a(n-1) - 12*(n-2)*a(n-2))/n, a(0)=1, a(1)=3.
%F E.g.f. (exp(6*x) + 1)/2 = 1 + 3*x/(G(0) - 6*x) where G(k) = 6*x + 1 + k - 6*x*(k+1)/G(k+1) (continued fraction, Euler's 1st kind, 1-step). (End)
%F "INVERT" transform of A000244. - _Alois P. Heinz_, Sep 22 2017
%e From _Adi Dani_, May 29 2011: (Start)
%e a(2)=18: there are 18 compositions of even natural numbers into 2 parts <= 5:
%e for 0: (0,0);
%e for 2: (0,2),(2,0),(1,1);
%e for 4: (0,4),(4,0),(1,3),(3,1),(2,2);
%e for 6: (1,5),(5,1),(2,4),(4,2),(3,3);
%e for 8: (3,5),(5,3),(4,4);
%e for 10: (5,5). (End)
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(3^j*a(n-j), j=1..n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 22 2017
%t Table[Ceiling[1/2(6^n)], {n, 0, 25}]
%t CoefficientList[Series[-(-1 + 3 x)/(1 - 6 x), {x, 0, 50}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 21 2011 *)
%t Join[{1},NestList[6#&,3,30]] (* _Harvey P. Dale_, May 25 2019 *)
%o (PARI) x='x+O('x^66); /* that many terms */
%o Vec((1-3*x)/(1-6*x)) /* show terms */ /* _Joerg Arndt_, May 29 2011 */
%Y Cf. A000244, A034494, A081340, A081342.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 18 2003
%E Typo in A-number fixed by _Klaus Brockhaus_, Apr 04 2010
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