This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004213 M3956 #81 Dec 16 2024 13:27:48
%S A004213 1,1,5,29,201,1657,15821,170389,2032785,26546673,376085653,5736591885,
%T A004213 93614616409,1625661357673,29905322979421,580513190237573,
%U A004213 11850869542405409,253669139947767777,5678266212792053029,132607996474971041789,3224106929536557918697
%N A004213 Shifts one place left under 4th-order binomial transform.
%C A004213 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
%D A004213 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A004213 Vincenzo Librandi, <a href="/A004213/b004213.txt">Table of n, a(n) for n = 0..66</a>
%H A004213 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 17.3.5, pp. 366-368
%H A004213 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. [Link to arXiv version]
%H A004213 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 A004213 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 A004213 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 A004213 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A004213 a(n) = Sum_{m=0..n} 4^(n-m)*Stirling2(n, m).
%F A004213 E.g.f.: exp((exp(4*x)-1)/4).
%F A004213 O.g.f. A(x) satisfies A'(x)/A(x) = e^(4x).
%F A004213 E.g.f.: exp(Integral_{t = 0..x} exp(4*t)). - _Joerg Arndt_, Apr 30 2011
%F A004213 O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-4*j*x). - _Joerg Arndt_, Apr 30 2011
%F A004213 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^(-1/4)*4^{n-1}*f_n(1/4). - _Milan Janjic_, May 30 2008
%F A004213 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:
%F A004213 1, 4, 0, 0, 0, ...
%F A004213 1, 1, 4, 0, 0, ...
%F A004213 1, 2, 1, 4, 0, ...
%F A004213 1, 3, 3, 1, 4, ...
%F A004213 ... - _Gary W. Adamson_, Jul 29 2011
%F A004213 G.f. satisfies A(x) = 1 + x/(1 - 4*x)*A(x/(1 - 4*x)). a(n) = Sum_{k = 1..n} 4^(n-k)*binomial(n-1,k-1)*a(k-1), n > 0, a(0) = 1. - _Vladimir Kruchinin_, Nov 28 2011 [corrected by _Ilya Gutkovskiy_, May 02 2019]
%F A004213 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
%F A004213 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
%F A004213 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
%F A004213 a(n) = exp(-1/4) * Sum_{k>=0} 4^(n-k) * k^n / k!. - _Vaclav Kotesovec_, Jul 15 2021
%F A004213 a(n) ~ 4^n * n^n * exp(n/LambertW(4*n) - 1/4 - n) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^n). - _Vaclav Kotesovec_, Jul 15 2021
%F A004213 From _Peter Bala_, Jun 29 2024: (Start)
%F A004213 a(n) = exp(-1/4)*Sum_{n >= 0} (4*n)^k/(n!*4^n).
%F A004213 Touchard's congruence holds: for odd prime p, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for odd prime p. See A004211. (End)
%e A004213 Restricted growth strings: a(0)=1 corresponds to the empty string, a(1)=1 to [0],
%e A004213 a(2)=3 to [00], [01], [02], [03], and [04], a(3) = 29 to
%e A004213 RGS F
%e A004213 .1: [ 0 0 0 ] [ 0 0 0 ]
%e A004213 .2: [ 0 0 1 ] [ 0 0 0 ]
%e A004213 .3: [ 0 0 2 ] [ 0 0 0 ]
%e A004213 .4: [ 0 0 3 ] [ 0 0 0 ]
%e A004213 .5: [ 0 0 4 ] [ 0 0 4 ]
%e A004213 .6: [ 0 1 0 ] [ 0 0 0 ]
%e A004213 .7: [ 0 1 1 ] [ 0 0 0 ]
%e A004213 .8: [ 0 1 2 ] [ 0 0 0 ]
%e A004213 .9: [ 0 1 3 ] [ 0 0 0 ]
%e A004213 10: [ 0 1 4 ] [ 0 0 4 ]
%e A004213 11: [ 0 2 0 ] [ 0 0 0 ]
%e A004213 12: [ 0 2 1 ] [ 0 0 0 ]
%e A004213 13: [ 0 2 2 ] [ 0 0 0 ]
%e A004213 14: [ 0 2 3 ] [ 0 0 0 ]
%e A004213 15: [ 0 2 4 ] [ 0 0 4 ]
%e A004213 16: [ 0 3 0 ] [ 0 0 0 ]
%e A004213 17: [ 0 3 1 ] [ 0 0 0 ]
%e A004213 18: [ 0 3 2 ] [ 0 0 0 ]
%e A004213 19: [ 0 3 3 ] [ 0 0 0 ]
%e A004213 20: [ 0 3 4 ] [ 0 0 4 ]
%e A004213 21: [ 0 4 0 ] [ 0 4 4 ]
%e A004213 22: [ 0 4 1 ] [ 0 4 4 ]
%e A004213 23: [ 0 4 2 ] [ 0 4 4 ]
%e A004213 24: [ 0 4 3 ] [ 0 4 4 ]
%e A004213 25: [ 0 4 4 ] [ 0 4 4 ]
%e A004213 26: [ 0 4 5 ] [ 0 4 4 ]
%e A004213 27: [ 0 4 6 ] [ 0 4 4 ]
%e A004213 28: [ 0 4 7 ] [ 0 4 4 ]
%e A004213 29: [ 0 4 8 ] [ 0 4 8 ]
%e A004213 [_Joerg Arndt_, Apr 30 2011]
%p A004213 A004213 := proc(n)
%p A004213 add(4^(n-m)*combinat[stirling2](n,m),m=0..n) ;
%p A004213 end proc:
%p A004213 seq(A004213(n),n=0..30) ; # _R. J. Mathar_, Aug 20 2022
%t A004213 Table[4^n BellB[n, 1/4], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)
%o A004213 (PARI) x='x+O('x^66);
%o A004213 egf=exp(intformal(exp(4*x))); /* = 1 + x + 5/2*x^2 + 29/6*x^3 + 67/8*x^4 + ... */
%o A004213 /* egf=exp(1/4*(exp(4*x)-1)) */ /* alternative computation */
%o A004213 Vec(serlaplace(egf)) /* _Joerg Arndt_, Apr 30 2011 */
%o A004213 (Maxima)
%o A004213 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 */
%Y A004213 Cf. A075499 (row sums).
%Y A004213 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
%K A004213 nonn,easy,eigen
%O A004213 0,3
%A A004213 _N. J. A. Sloane_