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 A088957 Hyperbinomial transform of the sequence of 1's. 7
 1, 2, 6, 29, 212, 2117, 26830, 412015, 7433032, 154076201, 3608522954, 94238893883, 2715385121740, 85574061070045, 2928110179818478, 108110945014584623, 4284188833355367440, 181370804507130015569, 8169524599872649117330, 390114757072969964280163 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See A088956 for the definition of the hyperbinomial transform. 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 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] LINKS Alois P. Heinz, Table of n, a(n) for n = 0..387 Marko Riedel et al., Proof of e.g.f. of sequence. FORMULA a(n) = Sum_{k=0..n} (n-k+1)^(n-k-1)*C(n, k). E.g.f.: A(x) = exp(x+sum(n>=1, n^(n-1)*x^n/n!)). E.g.f.: -LambertW(-x)*exp(x)/x. - Vladeta Jovovic, Oct 27 2003 a(n) ~ exp(1+exp(-1))*n^(n-1). - Vaclav Kotesovec, Jul 08 2013 EXAMPLE a(5) = 2117 = 1296 + 625 + 160 + 30 + 5 + 1 = sum of row 5 of triangle A088956. MAPLE a:= n-> add((n-j+1)^(n-j-1)*binomial(n, j), j=0..n): seq(a(n), n=0..20);  # Alois P. Heinz, Oct 30 2012 MATHEMATICA nn = 16; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[Series[Exp[x] Exp[t], {x, 0, nn}], x]  (* Geoffrey Critzer, Dec 29 2011 *) With[{nmax = 50}, CoefficientList[Series[-LambertW[-x]*Exp[x]/x, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *) PROG (Haskell) a088957 = sum . a088956_row  -- Reinhard Zumkeller, Jul 07 2013 (PARI) x='x+O('x^10); Vec(serlaplace(-lambertw(-x)*exp(x)/x)) \\ G. C. Greubel, Nov 14 2017 CROSSREFS Cf. A088956 (triangle). Row sums of A144289. - Alois P. Heinz, Jun 01 2009 Cf. A086331, A000169. Column k=1 of A144303. - Alois P. Heinz, Oct 30 2012 Sequence in context: A241462 A187006 A321961 * A030538 A326900 A181812 Adjacent sequences:  A088954 A088955 A088956 * A088958 A088959 A088960 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 26 2003 STATUS approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)