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A004212 Shifts one place left under 3rd order binomial transform.
(Formerly M3557)
12
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; text; internal format)
OFFSET

0,3

COMMENTS

Equals the eigensequence of triangle A027465, the cube of Pascal's triangle. [Gary W. Adamson, 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

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..200

Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368

I. M. Gessel, Applications of the classical umbral calculus. arXiv:math/0108121v3 [math.CO], 2001.

Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.

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, 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 Janjic, 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. [Vladimir Kruchinin, Nov 28 2011]

From Peter Bala, May 16 2012: (Start)

Recurrence equation: a(n+1) = sum {k = 0..n} 3^(n-k)*C(n,k)*a(k). Written umbrally this is a(n+1) = (a+3)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a+3) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a-3)*(a-6)*...*(a-3*(n-1)) = 1 with a(0) = 1. Compare with the Bell numbers B(n) = A000110(n), which satisfy the umbral recurrence B*(B-1)*...*(B-(n-1)) = 1 with B(0) = 1.

Touchard's congruence holds for prime p not equal to 3: a(p+k) = (a(k) + a(k+1)) (mod p) for k = 0,1,2,... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = 2 (mod p) for prime p <> 3.

(End)

G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-x*3*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

G.f.: T(0)/(1-x), where T(k) = 1 - 3*x^2*(k+1)/( 3*x^2*(k+1) - (1-x-3*x*k)*(1-4*x-3*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013

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

01:  [ 0 0 0 ]    [ 0 0 0 ]

02:  [ 0 0 1 ]    [ 0 0 0 ]

03:  [ 0 0 2 ]    [ 0 0 0 ]

04:  [ 0 0 3 ]    [ 0 0 3 ]

05:  [ 0 1 0 ]    [ 0 0 0 ]

06:  [ 0 1 1 ]    [ 0 0 0 ]

07:  [ 0 1 2 ]    [ 0 0 0 ]

08:  [ 0 1 3 ]    [ 0 0 3 ]

09:  [ 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]

MATHEMATICA

Table[Sum[StirlingS2[n, k] 3^(-k+n), {k, n}], {n, 20}] (* Vincenzo Librandi, May 21 2012 *)

Table[3^n BellB[n, 1/3], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

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, 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].

Cf. A009235.

Sequence in context: A241840 A199318 A117397 * A243241 A060905 A174123

Adjacent sequences:  A004209 A004210 A004211 * A004213 A004214 A004215

KEYWORD

nonn,easy,eigen

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 27 00:46 EDT 2017. Contains 285506 sequences.