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 A064618 #46 Jan 14 2022 11:48:24
%S A064618 1,1,5,49,821,21121,775205,38516689,2490976661,203419086241,
%T A064618 20474978755205,2490729330118129,360263844701062901,
%U A064618 61114158974786823361,12017074366801186956005,2711409826920884006692369,695820350706240448128979541,201526362605605903609254528481
%N A064618 Stirling transform of (n!)^2.
%C A064618 From _Thomas Wieder_, Oct 21 2004: (Start)
%C A064618 "Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
%C A064618 "Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
%C A064618 "E.g. for n=3 one has a(3) = 49 possible hierarchies:
%C A064618 "l_1:123,
%C A064618 "l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
%C A064618 "l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
%C A064618 "l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
%C A064618 "l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
%C A064618 "l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
%C A064618 "l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
%C A064618 "l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
%C A064618 "See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
%C A064618 (End)
%C A064618 Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - _Peter Bala_, Jan 15 2018
%H A064618 Alois P. Heinz, <a href="/A064618/b064618.txt">Table of n, a(n) for n = 0..253</a>
%F A064618 a(n) = Sum_{k=0..n} Stirling2(n, k)*(k!)^2.
%F A064618 E.g.f: hypergeom([1, 1], [], exp(x)-1). - _Vladeta Jovovic_, Sep 14 2003
%F A064618 O.g.f.: Sum_{n>=0} n!^2 * Product_{k=1..n} x/(1 - k*x). - _Paul D. Hanna_, Nov 25 2012
%F A064618 a(n) ~ exp(1/2) * (n!)^2. - _Vaclav Kotesovec_, May 10 2014
%p A064618 a:= n-> add(Stirling2(n, k)*(k!^2), k=0..n):
%p A064618 seq(a(n), n=0..20); # _Alois P. Heinz_, Apr 21 2012
%t A064618 Table[Sum[(k!)^2*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, May 10 2014 *)
%o A064618 (PARI) /* By Vladeta Jovovic's formula: */
%o A064618 {a(n) = my(X=x+x*O(x^n)); n!*polcoeff( sum(m=0,n, m!*(exp(X)-1)^m), n)} /* _Paul D. Hanna_, Feb 15 2012 */
%Y A064618 Cf. A001044.
%Y A064618 Cf. A001339.
%Y A064618 Cf. A242282, A242283, A019774, A320502.
%K A064618 nonn,easy
%O A064618 0,3
%A A064618 _Karol A. Penson_, Sep 26 2001