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A088957 Hyperbinomial transform of the sequence of 1's. 7

%I

%S 1,2,6,29,212,2117,26830,412015,7433032,154076201,3608522954,

%T 94238893883,2715385121740,85574061070045,2928110179818478,

%U 108110945014584623,4284188833355367440,181370804507130015569,8169524599872649117330,390114757072969964280163

%N Hyperbinomial transform of the sequence of 1's.

%C See A088956 for the definition of the hyperbinomial transform.

%C 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 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]

%H Alois P. Heinz, <a href="/A088957/b088957.txt">Table of n, a(n) for n = 0..387</a>

%H Marko Riedel et al., <a href="https://math.stackexchange.com/questions/3169295/">Proof of e.g.f. of sequence</a>.

%F a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k).

%F E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)).

%F E.g.f.: -LambertW(-x)*exp(x)/x. - _Vladeta Jovovic_, Oct 27 2003

%F a(n) ~ exp(1+exp(-1))*n^(n-1). - _Vaclav Kotesovec_, Jul 08 2013

%e a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956.

%p a:= n-> add((n-j+1)^(n-j-1)*binomial(n,j), j=0..n):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Oct 30 2012

%t nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];

%t Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x] (* _Geoffrey Critzer_, Dec 29 2011 *)

%t With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 14 2017 *)

%o (Haskell)

%o a088957 = sum . a088956_row -- _Reinhard Zumkeller_, Jul 07 2013

%o (PARI) x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ _G. C. Greubel_, Nov 14 2017

%Y Cf. A088956 (triangle).

%Y Row sums of A144289. - _Alois P. Heinz_, Jun 01 2009

%Y Cf. A086331, A000169.

%Y Column k=1 of A144303. - _Alois P. Heinz_, Oct 30 2012

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 26 2003

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Last modified February 20 08:05 EST 2020. Contains 332069 sequences. (Running on oeis4.)