A354556 Numerators of a sequence related to the Secretary Problem with Multiple Stoppings.

1, 3, 47, 2761, 4162637, 380537052235603, 705040594914523588948186792543, 302500210177484374840641189918370275991590974715547528765249, 49554292678269029432299170288905873298367846539726510384850403192729912522937262239403638817695466470734534217406992001

Offset: 1

Author

José María Grau Ribas, May 28 2022

Keywords

• nonn
• frac

Comments

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

Examples

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

Links

• T. Matsui and K. Ano, Lower Bounds for Bruss' Odds Problem with Multiple Stoppings, arXiv:1204.5537 [math.NT], 2012-2017.

Crossrefs

Cf. A354557 (denominators).

Programs

Mathematica

la[1]=1;alfa[k_,1]=1/k!;alfa[k_,k_]:=1;la[k_]:=la[k]=1-alfa[k,k-1];
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}];
S[n_]:=Sum[la[i],{i,1,n}];
Table[Numerator@S[n],{n,1,10}]