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A088957 Hyperbinomial transform of the sequence of 1's.

%I A088957 #63 Jan 29 2024 19:27:17
%S A088957 1,2,6,29,212,2117,26830,412015,7433032,154076201,3608522954,
%T A088957 94238893883,2715385121740,85574061070045,2928110179818478,
%U A088957 108110945014584623,4284188833355367440,181370804507130015569,8169524599872649117330,390114757072969964280163
%N A088957 Hyperbinomial transform of the sequence of 1's.
%C A088957 See A088956 for the definition of the hyperbinomial transform.
%C A088957 a(n) is the number of partial functions on {1,2,...,n} that are endofunctions with no cycles of length > 1.  The triangle A088956 classifies these functions according to the number of undefined elements in the domain.  The triangle A144289 classifies these functions according to the number of edges in their digraph representation (considering the empty function to have 1 edge).  The triangle A203092 classifies these functions according to the number of connected components. - _Geoffrey Critzer_, Dec 29 2011
%C A088957 a(n) is the number of rooted subtrees (for a fixed root) in the complete graph on n+1 vertices: a(3) = 29 is the number of rooted subtrees in K_4: 1 of size 1, 3 of size 2, 9 of size 3, and 16 spanning subtrees. - _Alex Chin_, Jul 25 2013 [corrected by _Marko Riedel_, Mar 31 2019]
%C A088957 From _Gus Wiseman_, Jan 28 2024: (Start)
%C A088957 Also the number of labeled loop-graphs on n vertices such that it is possible to choose a different vertex from each edge in exactly one way. For example, the a(3) = 29 uniquely choosable loop-graphs (loops shown as singletons) are:
%C A088957   {}  {1}  {1,2}  {1,12}   {1,2,13}  {1,12,13}
%C A088957       {2}  {1,3}  {1,13}   {1,2,23}  {1,12,23}
%C A088957       {3}  {2,3}  {2,12}   {1,3,12}  {1,13,23}
%C A088957                   {2,23}   {1,3,23}  {2,12,13}
%C A088957                   {3,13}   {2,3,12}  {2,12,23}
%C A088957                   {3,23}   {2,3,13}  {2,13,23}
%C A088957                   {1,2,3}            {3,12,13}
%C A088957                                      {3,12,23}
%C A088957                                      {3,13,23}
%C A088957 (End)
%H A088957 Alois P. Heinz, <a href="/A088957/b088957.txt">Table of n, a(n) for n = 0..387</a>
%H A088957 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/3169295/">Proof of e.g.f. of sequence</a>.
%F A088957 a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).
%F A088957 E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).
%F A088957 E.g.f.: -LambertW(-x)*exp(x)/x. - _Vladeta Jovovic_, Oct 27 2003
%F A088957 a(n) ~ exp(1+exp(-1))*n^(n-1). - _Vaclav Kotesovec_, Jul 08 2013
%F A088957 Binomial transform of A000272. - _Gus Wiseman_, Jan 25 2024
%e A088957 a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.
%p A088957 a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):
%p A088957 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 30 2012
%t A088957 nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
%t A088957 Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* _Geoffrey Critzer_, Dec 29 2011 *)
%t A088957 With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 14 2017 *)
%o A088957 (Haskell)
%o A088957 a088957 = sum . a088956_row  -- _Reinhard Zumkeller_, Jul 07 2013
%o A088957 (PARI) x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ _G. C. Greubel_, Nov 14 2017
%Y A088957 Cf. A088956 (triangle).
%Y A088957 Row sums of A144289. - _Alois P. Heinz_, Jun 01 2009
%Y A088957 Cf. A086331, A000169.
%Y A088957 Column k=1 of A144303. - _Alois P. Heinz_, Oct 30 2012
%Y A088957 The covering case is A000272, also the case of exactly n edges.
%Y A088957 Without the choice condition we have A006125 (shifted left).
%Y A088957 The unlabeled version is A087803.
%Y A088957 The choosable version is A368927, covering A369140, loopless A133686.
%Y A088957 The non-choosable version is A369141, covering A369142, loopless A367867.
%Y A088957 Cf. A000081, A000085, A057500, A062740, A137916, A277473, A322661, A367904, A368596, A368597, A368924.
%K A088957 nonn
%O A088957 0,2
%A A088957 _Paul D. Hanna_, Oct 26 2003