A080108 a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).

1, 2, 6, 23, 104, 537, 3100, 19693, 136064, 1013345, 8076644, 68486013, 614797936, 5818490641, 57846681092, 602259154853, 6548439927680, 74180742421185, 873588590481988, 10674437936521069, 135097459659312176

Offset: 1

Author

Vladeta Jovovic, Mar 15 2003

Keywords

• nonn

Comments

Row sums of triangle A154372. Example: a(3)=1+12+9+1=23. From A152818. - Paul Curtz, Jan 08 2009

Number of pointed set partitions of a pointed set k[1...k...n] with a prescribed point k. - Gus Wiseman, Sep 27 2015

With offset 0, a(n) is the number of partial functions (A000169) from [n]->[n] such that every element in the domain of definition is mapped to a fixed point. This implies a(n) is the number of idempotent partial functions Cf. A121337. - Geoffrey Critzer, Aug 07 2016

Examples

			G.f. = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + ...
The a(4) = 23 pointed set partitions of 1[1 2 3 4] are 1[1[1 2 3 4]], 1[1[1] 2[2 3 4]], 1[1[1] 3[2 3 4]], 1[1[1] 4[2 3 4]], 1[1[1 2] 3[3 4]], 1[1[1 2] 4[3 4]], 1[1[1 3] 2[2 4]], 1[1[1 3] 4[2 4]], 1[1[1 4] 2[2 3]], 1[1[1 4] 3[2 3]], 1[1[1 2 3] 4[4]], 1[1[1 2 4] 3[3]], 1[1[1 3 4] 2[2]], 1[1[1] 2[2] 3[3 4]], 1[1[1] 2[2] 4[3 4]], 1[1[1] 2[2 3] 4[4]], 1[1[1] 2[2 4] 3[3]], 1[1[1] 3[3] 4[2 4]], 1[1[1] 3[2 3] 4[4]], 1[1[1 2] 3[3] 4[4]], 1[1[1 3] 2[2] 4[4]], 1[1[1 4] 2[2] 3[3]], 1[1[1] 2[2] 3[3] 4[4]].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

First column of array A098697.

Cf. A152818, A185298, A262671.

Programs

Magma

[(1/n)*(&+[Binomial(n,k)*k^(n-k+1): k in [0..n]]): n in [1..30]]; // G. C. Greubel, Jan 22 2023

Mathematica

Table[Sum[k^(n-k) Binomial[n-1,k-1],{k,n}],{n,30}] (* Harvey P. Dale, Aug 19 2012 *)
Table[SeriesCoefficient[Sum[x^k/(1-k*x)^k,{k,0,n}],{x,0,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 06 2014 *)
CoefficientList[Series[E^(x*(1+E^x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 06 2014 *)

PARI

a(n)=sum(k=1,n, k^(n-k)*binomial(n-1,k-1)) \\ Anders Hellström, Sep 27 2015

SageMath

def A080108(n): return (1/n)*sum(binomial(n,k)*k^(n-k+1) for k in range(n+1))
[A080108(n) for n in range(1,31)] # G. C. Greubel, Jan 22 2023

Formula

G.f.: Sum_{k>0} x^k/(1-k*x)^k.

E.g.f. (for offset 0): exp(x*(1+exp(x))). - Vladeta Jovovic, Aug 25 2003

a(n) = A185298(n)/n.