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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 January 22 10:25 EST 2020. Contains 331144 sequences. (Running on oeis4.)