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A004213 Shifts one place left under 4th order binomial transform.
(Formerly M3956)
9
1, 1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, 1625661357673, 29905322979421, 580513190237573, 11850869542405409, 253669139947767777, 5678266212792053029, 132607996474971041789, 3224106929536557918697 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+4 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=4, 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..66

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

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

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((4^(n-m))*stirling2(n, m), m=0..n).

E.g.f.: exp((exp(4*x)-1)/4).

O.g.f. A(x) satisfies A'(x)/A(x) = e^(4x).

E.g.f.: exp(int(t=0..x, exp(4*t))). [Joerg Arndt, Apr 30 2011]

O.g.f.: sum(k>=0, x^k/prod(j=1..k, (1-4*j*x))). [Joerg Arndt, Apr 30 2011]

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^{-1/4}*4^{n-1}*f_n(1/4). - Milan Janjic, May 30 2008

a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (4,4,4,...) is appended to the right of Pascal's triangle:

1, 4, 0, 0, 0,...

1, 1, 4, 0, 0,...

1, 2, 1, 4, 0,...

1, 3, 3, 1, 4,...

... - Gary W. Adamson, Jul 29 2011

G.f. satisfies A(x)=1+x/(1-4*x)*A(4*x/(1-4*x)). a(n)=sum(4^(n-k)*binomial(n-1,k-1)*a(k-1),k,1,n), n>0, a(0)=1. [Vladimir Kruchinin, Nov 28 2011]

G.f.: (G(0) - 1)/(x-1) where G(k) =  1 - 1/(1-4*k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 24 2013

G.f.: (G(0) - 1)/(1+x) where G(k) =  1 + 1/(1-4*k*x)/(1-x/(x+1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 31 2013

G.f.: T(0)/(1-x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-x-4*x*k)*(1-5*x-4*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], [03], and [04], a(3) = 29 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 0 ]

.5:  [ 0 0 4 ]    [ 0 0 4 ]

.6:  [ 0 1 0 ]    [ 0 0 0 ]

.7:  [ 0 1 1 ]    [ 0 0 0 ]

.8:  [ 0 1 2 ]    [ 0 0 0 ]

.9:  [ 0 1 3 ]    [ 0 0 0 ]

10:  [ 0 1 4 ]    [ 0 0 4 ]

11:  [ 0 2 0 ]    [ 0 0 0 ]

12:  [ 0 2 1 ]    [ 0 0 0 ]

13:  [ 0 2 2 ]    [ 0 0 0 ]

14:  [ 0 2 3 ]    [ 0 0 0 ]

15:  [ 0 2 4 ]    [ 0 0 4 ]

16:  [ 0 3 0 ]    [ 0 0 0 ]

17:  [ 0 3 1 ]    [ 0 0 0 ]

18:  [ 0 3 2 ]    [ 0 0 0 ]

19:  [ 0 3 3 ]    [ 0 0 0 ]

20:  [ 0 3 4 ]    [ 0 0 4 ]

21:  [ 0 4 0 ]    [ 0 4 4 ]

22:  [ 0 4 1 ]    [ 0 4 4 ]

23:  [ 0 4 2 ]    [ 0 4 4 ]

24:  [ 0 4 3 ]    [ 0 4 4 ]

25:  [ 0 4 4 ]    [ 0 4 4 ]

26:  [ 0 4 5 ]    [ 0 4 4 ]

27:  [ 0 4 6 ]    [ 0 4 4 ]

28:  [ 0 4 7 ]    [ 0 4 4 ]

29:  [ 0 4 8 ]    [ 0 4 8 ]

[Joerg Arndt, Apr 30 2011]

MATHEMATICA

Table[4^n BellB[n, 1/4], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

PROG

(PARI) x='x+O('x^66);

egf=exp(intformal(exp(4*x))); /* =  1 + x + 5/2*x^2 + 29/6*x^3 + 67/8*x^4 + ... */

/* egf=exp(1/4*(exp(4*x)-1)) */ /* alternative computation */

Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

(Maxima)

a(n):=if n=0 then 1 else sum(4^(n-k)*binomial(n-1, k-1)*a(k-1), k, 1, n); \\ Vladimir Kruchinin, Nov 28 2011

CROSSREFS

Cf. A075499 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A005011 (s(k)<=F(k)+5), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]

Sequence in context: A216313 A108453 A201203 * A105277 A103213 A057588

Adjacent sequences:  A004210 A004211 A004212 * A004214 A004215 A004216

KEYWORD

nonn,easy,eigen

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 23 02:54 EDT 2017. Contains 288633 sequences.