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

0,2

Comments

Binomial transform of A081340. 3rd binomial transform of (1,0,9,0,81,0,729,0,...).

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

Number of compositions of even natural numbers in n parts <= 5. - Adi Dani, May 29 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..125

Index entries for linear recurrences with constant coefficients, signature (6).

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) = A000244(n)*A011782(n). - Philippe Deléham, Dec 01 2008

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

Example

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)

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 *)

Prog

(PARI) x='x+O('x^66); /* that many terms */
Vec((1-3*x)/(1-6*x)) /* show terms */ /* Joerg Arndt, May 29 2011 */

Crossrefs

Cf. A000244, A034494, A081340, A081342.

Sequence in context: A137962 A267662 A169604 * A355353 A363439 A132900

Adjacent sequences: A081338 A081339 A081340 * A081342 A081343 A081344

Keyword

easy,nonn

Author

Paul Barry, Mar 18 2003

Extensions

Typo in A-number fixed by Klaus Brockhaus, Apr 04 2010

Status

approved