cp's OEIS Frontend

A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.

%I A322380 #41 Feb 24 2022 18:22:36
%S A322380 1,1,1,5,7,37,79,173,101,127,1033,1571,200069,2564519,5126711,
%T A322380 25661369,532393,431100529,1855391,1533985991,48977868113,
%U A322380 342880481117,342289639579,435979161889,1308720597671,373092965489,7824703695283,24141028973,31250466692609
%N A322380 Numerator of the sum of inverse products of parts in all strict partitions of n.
%C A322380 a(n)/A322381(n) = A007838(n)/A000142(n) is the probability that a random permutation of [n] has distinct cycle sizes. - _Geoffrey Critzer_, Feb 23 2022
%H A322380 Alois P. Heinz, <a href="/A322380/b322380.txt">Table of n, a(n) for n = 0..1268</a>
%H A322380 Andreas B. G. Blobel, <a href="https://arxiv.org/abs/1904.07808">An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k)</a>, arXiv:1904.07808 [math.CO], 2019.
%H A322380 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 137.
%H A322380 A. Knopfmacher and J. N. Ridley, <a href="http://dx.doi.org/10.1137/0406031">Reciprocal sums over partitions and compositions</a>, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
%H A322380 D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21123.pdf">On reciprocally weighted partitions</a>, Acta Arithmetica XXI (1972), 379-388.
%H A322380 D. Zeilberger, N. Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/fcp.html">Fractional Counting of Integer Partitions</a>, 2018.
%F A322380 Limit_{n->infinity} a(n)/A322381(n) = exp(-gamma) = A080130.
%F A322380 Sum_{n>=0} a(n)/A322381(n)*x^n = Product_{i>=1} (1 + x^i/i). - _Geoffrey Critzer_, Feb 23 2022
%e A322380 1/1, 1/1, 1/2, 5/6, 7/12, 37/60, 79/120, 173/280, 101/168, 127/210, 1033/1680, 1571/2640, 200069/332640, 2564519/4324320, 5126711/8648640, ... = A322380/A322381
%p A322380 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A322380       b(n, i-1) +b(n-i, min(i-1, n-i))/i))
%p A322380     end:
%p A322380 a:= n-> numer(b(n$2)):
%p A322380 seq(a(n), n=0..30);
%t A322380 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
%t A322380 a[n_] := Numerator[b[n, n]];
%t A322380 a /@ Range[0, 30] (* _Jean-François Alcover_, Feb 25 2020, after _Alois P. Heinz_ *)
%Y A322380 Denominators: A322381.
%Y A322380 Cf. A000009, A000142, A007838, A022629, A080130, A177208, A177209, A322364, A322365, A323290, A323291, A323339, A323340.
%K A322380 nonn,frac
%O A322380 0,4
%A A322380 _Alois P. Heinz_, Dec 05 2018