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 A005012 M4434 #64 Feb 16 2025 08:32:28
%S A005012 1,1,7,55,505,5497,69823,1007407,16157905,284214097,5432922775,
%T A005012 112034017735,2476196276617,58332035387017,1457666574447247,
%U A005012 38485034941511935,1069787864231083297,31213730550761076769,953352927192964243879,30406448846308128741847
%N A005012 Shifts one place left under 6th-order binomial transform.
%D A005012 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005012 Robert Israel, <a href="/A005012/b005012.txt">Table of n, a(n) for n = 0..420</a>
%H A005012 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A005012 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 A005012 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 A005012 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 A005012 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A005012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell polynomial</a>
%F A005012 a(n) = sum((6^(n-m))*stirling2(n,m), m=0..n). stirling2(n,m)=A008277(n,m).
%F A005012 E.g.f.: exp((exp(6*x)-1)/6) satisfies A'(x)/A(x) = exp(6*x).
%F A005012 G.f.: T(0)/(x*(1-x)) -1/x, where T(k) = 1 - 6*x^2*(k+1)/( 6*x^2*(k+1) - (1-x-6*x*k)*(1-7*x-6*x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 25 2013
%F A005012 a(n) = 6^n * B(n,1/6) where B(n,x) is the Bell polynomial of degree n. - _Vladimir Reshetnikov_, Oct 20 2015
%F A005012 O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 6*j*x). - _Ilya Gutkovskiy_, Mar 20 2018
%F A005012 a(n) ~ 6^n * n^n * exp(n/LambertW(6*n) - 1/6 - n) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^n). - _Vaclav Kotesovec_, Jul 15 2021
%p A005012 seq(6^n*BellB(n,1/6), n = 0 .. 50); # _Robert Israel_, Oct 20 2015
%t A005012 Table[6^n BellB[n, 1/6], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)
%o A005012 (GAP) List([0..20],n->Sum([0..n],m->6^(n-m)*Stirling2(n,m))); # _Muniru A Asiru_, Mar 20 2018
%Y A005012 Cf. A004211.
%K A005012 nonn,easy,eigen
%O A005012 0,3
%A A005012 _N. J. A. Sloane_, _Simon Plouffe_
%E A005012 a(0)=1 inserted by _Alois P. Heinz_, Oct 20 2015