A005011 Shifts one place left under 5th-order binomial transform. (Formerly M4240)

1, 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, 11458179765541, 249255304141006, 5725640423174901, 138407987170952351, 3510263847256823056

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, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.

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.

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

N. J. A. Sloane, Transforms

Formula

a(n) = Sum_{m=0..n} 5^(n-m)*Stirling2(n, m), 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(Integral_{t = 0..x} exp(5*t)). - Joerg Arndt, Apr 30 2011

O.g.f.: Sum_{k>=0} x^k/Product_{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) = (d/dx)(x*f_n(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

G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 - 5*x))/(1 - 5*x). - Ilya Gutkovskiy, May 03 2019

a(n) ~ 5^n * n^n * exp(n/LambertW(5*n) - 1/5 - n) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^n). - Vaclav Kotesovec, Jul 15 2021

From Peter Bala, Jun 29 2024: (Start)

a(n) = exp(-1/5)*Sum_{n >= 0} (5*n)^k/(n!*5^n).

Touchard's congruence holds: for all primes p != 5, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for all primes p != 5. See A004211. (End)

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: A307663 A345189 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