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 A102262 Numerators of probabilities in gift exchange problem with n people. 3
 0, 1, 5, 19, 203, 4343, 63853, 58129, 160127, 8885501, 1500518539, 404156337271, 16040576541971, 1694200740145637, 24047240650458731, 22823917472900053, 2511014355032164231, 143734030512459889193, 49611557898193759558813, 950601970122346247310883 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,3 COMMENTS This is a version of the Secret Santa game. n friends organize a gift exchange. The n names are put into a hat and the first person draws one. If she picks her own name, then she returns it to the bag and draws again, repeating until she has a name that is not her own. Then the second person draws, again returning his own name if it is drawn. This continues down the line. What is the probability p(n) that when the n-th person draws, only her own name will be left in the bag? I heard about the problem from Gary Thompson at Grove City College in PA. LINKS Jon E. Schoenfield, Table of n, a(n) for n = 2..389 Math Forum at Drexel, A variant on the "Secret Santa" FORMULA From Jon E. Schoenfield, Sep 30 2006: (Start) p(n) = Sum_{i=1..n-2} t(n,i)/(n-1)!^2 where   t(n,i) = (n-2)*i^2/(i-1)*t(n-1,i-1) - (n-i-2)*t(n-1,i) for 1 < i < n-1;   t(n,1) = (-1)^(n-1)*(n-1)!/2 for i = 1 and n > 2;   t(n,i) = 0 otherwise. (End) Based on the values of p(n) for n <= 1000, it seems plausible that, as n increases, p(n) approaches 1/(n + log(n) + EulerGamma), where EulerGamma = 0.5772156649015... (the Euler-Mascheroni constant). - Jon E. Schoenfield, Dec 11 2021 EXAMPLE p(2) through p(10) are 0, 1/4, 5/36, 19/144, 203/1800, 4343/43200, 63853/705600, 58129/705600, 160127/2116800. PROG (Magma) N:=21; a:=[]; row:=[]; T:=[]; for n in [2..N] do row[n-1]:=0; T[n]:=row; T[n]:=(-1)^(n-1)*Factorial(n-1) div 2; for i in [2..n-2] do T[n][i]:=(n-2)*i^2/(i-1)*T[n-1][i-1]-(n-i-2)*T[n-1][i]; end for; p:=0; for i in [1..n-2] do p+:=T[n][i]/Factorial(n-1)^2; end for; a[#a+1]:=Numerator(p); end for; a; // Jon E. Schoenfield, Dec 10 2021 CROSSREFS Cf. A102263, A136300. Sequence in context: A145935 A024529 A106991 * A123281 A135171 A058765 Adjacent sequences:  A102259 A102260 A102261 * A102263 A102264 A102265 KEYWORD nonn,frac AUTHOR Jerrold Grossman, Feb 17 2005 EXTENSIONS More terms from Jon E. Schoenfield, Sep 30 2006 STATUS approved

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Last modified September 27 18:50 EDT 2022. Contains 357062 sequences. (Running on oeis4.)