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A081341 Expansion of exp(3*x)*cosh(3*x). 13
1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568 (list; graph; refs; listen; history; text; internal format)
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<=k<=n} A134309(n,k)*3^k = Sum_{k, 0<=k<=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 *)

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 * A132900 A050623 A037760

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

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Last modified June 22 17:19 EDT 2018. Contains 305672 sequences. (Running on oeis4.)