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%I #52 Nov 26 2024 03:35:06
%S 1,2,6,23,104,537,3100,19693,136064,1013345,8076644,68486013,
%T 614797936,5818490641,57846681092,602259154853,6548439927680,
%U 74180742421185,873588590481988,10674437936521069,135097459659312176
%N a(n) = Sum_{k=1..n} k^(n-k)*binomial(n-1,k-1).
%C Row sums of triangle A154372. Example: a(3)=1+12+9+1=23. From A152818. - _Paul Curtz_, Jan 08 2009
%C Number of pointed set partitions of a pointed set k[1...k...n] with a prescribed point k. - _Gus Wiseman_, Sep 27 2015
%C 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
%H Vincenzo Librandi, <a href="/A080108/b080108.txt">Table of n, a(n) for n = 1..200</a>
%F G.f.: Sum_{k>0} x^k/(1-k*x)^k.
%F E.g.f. (for offset 0): exp(x*(1+exp(x))). - _Vladeta Jovovic_, Aug 25 2003
%F a(n) = A185298(n)/n.
%e G.f. = x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + ...
%e 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]].
%t Table[Sum[k^(n-k) Binomial[n-1,k-1],{k,n}],{n,30}] (* _Harvey P. Dale_, Aug 19 2012 *)
%t Table[SeriesCoefficient[Sum[x^k/(1-k*x)^k,{k,0,n}],{x,0,n}], {n,1,20}] (* _Vaclav Kotesovec_, Aug 06 2014 *)
%t CoefficientList[Series[E^(x*(1+E^x)), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 06 2014 *)
%o (PARI) a(n)=sum(k=1,n, k^(n-k)*binomial(n-1,k-1)) \\ _Anders Hellström_, Sep 27 2015
%o (Magma) [(1/n)*(&+[Binomial(n,k)*k^(n-k+1): k in [0..n]]): n in [1..30]]; // _G. C. Greubel_, Jan 22 2023
%o (SageMath)
%o def A080108(n): return (1/n)*sum(binomial(n,k)*k^(n-k+1) for k in range(n+1))
%o [A080108(n) for n in range(1,31)] # _G. C. Greubel_, Jan 22 2023
%Y First column of array A098697.
%Y Cf. A152818, A185298, A262671.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Mar 15 2003