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Expansion of exp(3*x)*cosh(3*x).
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%I #55 Aug 23 2024 20:57:48

%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