A081341 Expansion of exp(3*x)*cosh(3*x).
1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568
Offset: 0
Examples
From _Adi Dani_, May 29 2011: (Start) a(2)=18: there are 18 compositions of even natural numbers into 2 parts <= 5: for 0: (0,0); for 2: (0,2),(2,0),(1,1); for 4: (0,4),(4,0),(1,3),(3,1),(2,2); for 6: (1,5),(5,1),(2,4),(4,2),(3,3); for 8: (3,5),(5,3),(4,4); for 10: (5,5). (End)
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..125
- Index entries for linear recurrences with constant coefficients, signature (6).
Programs
-
Maple
a:= proc(n) option remember; `if`(n=0, 1, add(3^j*a(n-j), j=1..n)) end: seq(a(n), n=0..30); # Alois P. Heinz, Sep 22 2017 -
Mathematica
Table[Ceiling[1/2(6^n)], {n, 0, 25}] CoefficientList[Series[-(-1 + 3 x)/(1 - 6 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *) Join[{1},NestList[6#&,3,30]] (* Harvey P. Dale, May 25 2019 *) -
PARI
x='x+O('x^66); /* that many terms */ Vec((1-3*x)/(1-6*x)) /* show terms */ /* Joerg Arndt, May 29 2011 */
Formula
a(0)=1, a(n) = 6^n/2, n > 0.
G.f.: (1-3*x)/(1-6*x).
E.g.f.: exp(3*x)*cosh(3*x).
a(n) = ((3+sqrt(9))^n + (3-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
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
From Sergei N. Gladkovskii, Jul 19 2012: (Start)
a(n) = ((8*n-4)*a(n-1) - 12*(n-2)*a(n-2))/n, a(0)=1, a(1)=3.
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)
"INVERT" transform of A000244. - Alois P. Heinz, Sep 22 2017
Extensions
Typo in A-number fixed by Klaus Brockhaus, Apr 04 2010
Comments