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 A368725 #29 Jul 18 2025 14:44:41
%S A368725 1,1,2,9,100,1065,10626,224161,4598504,46288017,2509940710,
%T A368725 84061763841,-1602021820596,164372205860473,5216105126641514,
%U A368725 -883395389739028095,79008645559978113616,-1023235751229436800735,-651030746777115881959602,113943411938145511923004513
%N A368725 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^n / k!.
%H A368725 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A368725 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>.
%F A368725 E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 + k * x)).
%F A368725 a(n) = n! * [x^n] B_n(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
%F A368725 a(n) ~ A000587(n) * (-1)^n * exp(-1) * n!. - _Vaclav Kotesovec_, Jul 18 2025
%t A368725 Join[{1}, Table[n!*Sum[(-1)^(n-k)*k^n/k!, {k, 0, n}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Jul 18 2025 *)
%o A368725 (PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*k^n/k!);
%Y A368725 Main diagonal of A368724.
%Y A368725 Cf. A000587, A256016.
%K A368725 sign
%O A368725 0,3
%A A368725 _Seiichi Manyama_, Jan 04 2024