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A005011 Shifts one place left under 5th order binomial transform.
(Formerly M4240)
12
1, 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, 11458179765541, 249255304141006, 5725640423174901, 138407987170952351, 3510263847256823056 (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)+5 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=5, otherwise F(k+1)=F(k); see examples in A004211, A004212, and A004213, 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.

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

E.g.f.: exp((exp(5*x)-1)/5).

O.g.f. A(x) satisfies A'(x)/A(x) = exp(5*x).

E.g.f.: exp(int(t=0..x, exp(5*t))). [Joerg Arndt, Apr 30 2011]

O.g.f.: sum(k>=0, x^k/prod(j=1..k, (1-5*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^{-1/5}*5^{n-1}*f_n(1/5). - Milan Janjic, May 30 2008

a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (5,5,5,...) is appended to the right of Pascal's triangle:

1, 5, 0, 0, 0,...

1, 1, 5, 0, 0,...

1, 2, 1, 5, 0,...

1, 3, 3, 1, 5,...

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

G.f.: T(0)/(1-x), where T(k) = 1 - 5*x^2*(k+1)/( 5*x^2*(k+1) - (1-x-5*x*k)*(1-6*x-5*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013

MATHEMATICA

Table[5^n BellB[n, 1/5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

PROG

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

egf=exp(intformal(exp(5*x))); /* =  1 + x + 3*x^2 + 41/6*x^3 + 331/24*x^4 + ... */

/* egf=exp(1/5*(exp(5*x)-1)) */ /* alternative computation */

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

CROSSREFS

Cf. A075500 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A004213 (s(k)<=F(k)+4), A000110 (s(k)<=F(k)+1) [Joerg Arndt, Apr 30 2011]

Sequence in context: A199553 A225031 A083430 * A187814 A009122 A184140

Adjacent sequences:  A005008 A005009 A005010 * A005012 A005013 A005014

KEYWORD

easy,nonn,eigen

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.