cp's OEIS Frontend

A005011 Shifts one place left under 5th-order binomial transform.

%I A005011 M4240 #66 Jul 01 2024 13:15:30
%S A005011 1,1,6,41,331,3176,35451,447981,6282416,96546231,1611270851,
%T A005011 28985293526,558413253581,11458179765541,249255304141006,
%U A005011 5725640423174901,138407987170952351,3510263847256823056
%N A005011 Shifts one place left under 5th-order binomial transform.
%C A005011 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]
%D A005011 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005011 Vincenzo Librandi, <a href="/A005011/b005011.txt">Table of n, a(n) for n = 0..66</a>
%H A005011 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 17.3.5, pp. 366-368
%H A005011 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A005011 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A005011 Adalbert Kerber, <a href="http://dx.doi.org/10.1016/0012-365X(78)90163-2">A matrix of combinatorial numbers related to the symmetric groups</a>, Discrete Math., 21 (1978), 319-321.
%H A005011 A. Kerber, <a href="/A004211/a004211.pdf">A matrix of combinatorial numbers related to the symmetric groups<</a>, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
%H A005011 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A005011 a(n) = Sum_{m=0..n} 5^(n-m)*Stirling2(n, m), n >= 0.
%F A005011 E.g.f.: exp((exp(5*x)-1)/5).
%F A005011 O.g.f. A(x) satisfies A'(x)/A(x) = exp(5*x).
%F A005011 E.g.f.: exp(Integral_{t = 0..x} exp(5*t)). - _Joerg Arndt_, Apr 30 2011
%F A005011 O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-5*j*x). - _Joerg Arndt_, Apr 30 2011
%F A005011 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
%F A005011 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:
%F A005011   1, 5, 0, 0, 0, ...
%F A005011   1, 1, 5, 0, 0, ...
%F A005011   1, 2, 1, 5, 0, ...
%F A005011   1, 3, 3, 1, 5, ...
%F A005011   ... - _Gary W. Adamson_, Jul 29 2011
%F A005011 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
%F A005011 G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 - 5*x))/(1 - 5*x). - _Ilya Gutkovskiy_, May 03 2019
%F A005011 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
%F A005011 From _Peter Bala_, Jun 29 2024: (Start)
%F A005011 a(n) = exp(-1/5)*Sum_{n >= 0} (5*n)^k/(n!*5^n).
%F A005011 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)
%t A005011 Table[5^n BellB[n, 1/5], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)
%o A005011 (PARI) x='x+O('x^66);
%o A005011 egf=exp(intformal(exp(5*x))); /* =  1 + x + 3*x^2 + 41/6*x^3 + 331/24*x^4 + ... */
%o A005011 /* egf=exp(1/5*(exp(5*x)-1)) */ /* alternative computation */
%o A005011 Vec(serlaplace(egf)) /* _Joerg Arndt_, Apr 30 2011 */
%Y A005011 Cf. A075500 (row sums).
%Y A005011 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
%K A005011 easy,nonn,eigen
%O A005011 0,3
%A A005011 _N. J. A. Sloane_, _Simon Plouffe_