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Numerators of a sequence related to the Secretary Problem with Multiple Stoppings.
2

%I #22 Aug 01 2022 08:11:18

%S 1,3,47,2761,4162637,380537052235603,705040594914523588948186792543,

%T 302500210177484374840641189918370275991590974715547528765249,

%U 49554292678269029432299170288905873298367846539726510384850403192729912522937262239403638817695466470734534217406992001

%N Numerators of a sequence related to the Secretary Problem with Multiple Stoppings.

%C exp(-A354556(n)/A354557(n)) is the asymptotic value of the n-th threshold in the Secretary Problem with n Stoppings.

%H T. Matsui and K. Ano, <a href="https://arxiv.org/abs/1204.5537">Lower Bounds for Bruss' Odds Problem with Multiple Stoppings</a>, arXiv:1204.5537 [math.NT], 2012-2017.

%e 1, 3/2, 47/24, 2761/1152, 4162637/1474560, 380537052235603/117413668454400, 705040594914523588948186792543/193003573558876719588311040000

%t la[1]=1;alfa[k_,1]=1/k!;alfa[k_,k_]:=1;la[k_]:=la[k]=1-alfa[k,k-1];

%t alfa[k_,kp_]:=alfa[k,kp]= la[kp]^(1+k-kp)/(1+k-kp)!+Sum[la[kp]^b/b! alfa[k-b,kp-1],{b,0,k-kp}];

%t S[n_]:=Sum[la[i],{i,1,n}];

%t Table[Numerator@S[n],{n,1,10}]

%Y Cf. A354557 (denominators).

%K nonn,frac

%O 1,2

%A _José María Grau Ribas_, May 28 2022