|
| |
|
|
A004212
|
|
Shifts one place left under 3rd order binomial transform.
(Formerly M3557)
|
|
5
| |
|
|
1, 1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, 140329768789, 2087182244308, 32725315072135, 539118388883449, 9304591246975030, 167804098493079547, 3155000165773280893
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Equals the eigensequence of triangle A027465, the cube of Pascal's triangle. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+3 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=3, otherwise F(k+1)=F(k); see example and Fxtbook link. [Joerg Arndt, Apr 30 2011]
|
|
|
REFERENCES
| A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..66
Joerg Arndt, Fxtbook, section 17.3.5, pp. 366-368
N. J. A. Sloane, Transforms
|
|
|
FORMULA
| a_n=sum(3^(n-k)*stirling2(n, k), k=0..n). - Emeric Deutsch, Feb 11, 2002
E.g.f.: exp((exp(3*x)-1)/3).
O.g.f. A(x) satisfies A'(x)/A(x) = e^(3*x).
E.g.f.: exp(int(t=0..x, exp(3*t))). [Joerg Arndt, Apr 30 2011]
O.g.f.: sum(k>=0, x^k/prod(j=1..k, (1-3*j*x))). [Joerg Arndt, Apr 30 2011]
Hankel transform is A000178(n)*3^C(n+1,2). - Paul Barry (pbarry(AT)wit.ie), Mar 31 2008
Define f_1(x),f_2(x),... such that f_1(x)=e^x, f_{n+1}(x)=diff(x*f_n(x),x), for n=2,3,.... Then a(n)=e^{-1/2}*3^{n-1}*f_n(1/3). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008
a(n) = the upper left term in M^n, M = the following infinite square production matrix:
1, 3, 0, 0, 0,...
1, 1, 3, 0, 0,...
1, 2, 1, 3, 0,...
1, 3, 3, 1, 3,...
...(in which a diagonal of (3,3,3,...) is appended to the right of Pascal's triangle). - Gary W. Adamson, Jul 29 2011
G.f. satisfies A(x)=1+x/(1-3*x)*A(3*x/(1-3*x)). a(n)=sum(3^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n), n>0, a(0)=1. [ From Vladimir Kruchinin, Nov 28 2011]
|
|
|
EXAMPLE
| Restricted growth strings: a(0)=1 corresponds to the empty string, a(1)=1 to [0],
a(2)=3 to [00], [01], [02], and [03], a(3) = 19 to
RGS F
1: [ 0 0 0 ] [ 0 0 0 ]
2: [ 0 0 1 ] [ 0 0 0 ]
3: [ 0 0 2 ] [ 0 0 0 ]
4: [ 0 0 3 ] [ 0 0 3 ]
5: [ 0 1 0 ] [ 0 0 0 ]
6: [ 0 1 1 ] [ 0 0 0 ]
7: [ 0 1 2 ] [ 0 0 0 ]
8: [ 0 1 3 ] [ 0 0 3 ]
9: [ 0 2 0 ] [ 0 0 0 ]
10: [ 0 2 1 ] [ 0 0 0 ]
11: [ 0 2 2 ] [ 0 0 0 ]
12: [ 0 2 3 ] [ 0 0 3 ]
13: [ 0 3 0 ] [ 0 3 3 ]
14: [ 0 3 1 ] [ 0 3 3 ]
15: [ 0 3 2 ] [ 0 3 3 ]
16: [ 0 3 3 ] [ 0 3 3 ]
17: [ 0 3 4 ] [ 0 3 3 ]
18: [ 0 3 5 ] [ 0 3 3 ]
19: [ 0 3 6 ] [ 0 3 6 ]
[Joerg Arndt, Apr 30 2011]
|
|
|
PROG
| (Pari) x='x+O('x^66); /* that many terms */
egf=exp(intformal(exp(3*x))); /* = 1 + x + 2*x^2 + 19/6*x^3 + 109/24*x^4 + ... */
/* egf=exp(1/3*(exp(3*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */
(Maxima)
a(n):=if n=0 then 1 else sum(3^(n-k)*binomial(n-1, k-1)*a(k-1), k, 1, n); [From Vladimir Kruchinin, Nov 28 2011]
|
|
|
CROSSREFS
| Cf. A075498 (row sums).
A027465 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
A004211 (RGS where s(k)<=F(k)+2), A004213 (s(k)<=F(k)+4), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]
Sequence in context: A091643 A199318 A117397 * A060905 A174123 A127548
Adjacent sequences: A004209 A004210 A004211 * A004213 A004214 A004215
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
