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A322364 Numerator of the sum of inverse products of parts in all partitions of n.

%I A322364 #26 Apr 29 2020 07:44:31
%S A322364 1,1,3,11,7,27,581,4583,2327,69761,775643,147941,30601201,30679433,
%T A322364 10928023,6516099439,445868889691,298288331489,7327135996801,
%U A322364 1029216937671847,14361631943741,837902013393451,2766939485246012129,274082602410356881,835547516381094139939
%N A322364 Numerator of the sum of inverse products of parts in all partitions of n.
%H A322364 Alois P. Heinz, <a href="/A322364/b322364.txt">Table of n, a(n) for n = 0..505</a>
%H A322364 A. Knopfmacher, 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 A322364 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 A322364 D. Zeilberger, N. Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/fcp.html">Fractional Counting of Integer Partitions</a>, 2018.
%F A322364 Limit_{n-> infinity} a(n)/(n*A322365(n)) = exp(-gamma) = A080130.
%e A322364 1/1, 1/1, 3/2, 11/6, 7/3, 27/10, 581/180, 4583/1260, 2327/560, 69761/15120, 775643/151200, 147941/26400, 30601201/4989600, 30679433/4633200 ... = A322364/A322365
%p A322364 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A322364       b(n, i-1) +b(n-i, min(i, n-i))/i)
%p A322364     end:
%p A322364 a:= n-> numer(b(n$2)):
%p A322364 seq(a(n), n=0..30);
%t A322364 b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
%t A322364 a[n_] := Numerator[b[n, n]];
%t A322364 a /@ Range[0, 30] (* _Jean-François Alcover_, Apr 29 2020, after _Alois P. Heinz_ *)
%o A322364 (PARI) a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s);} \\ _Michel Marcus_, Apr 29 2020
%Y A322364 Denominators: A322365.
%Y A322364 Cf. A000041, A006906, A080130, A177208, A177209, A322380, A322381, A323290, A323291, A323339, A323340.
%K A322364 nonn,frac
%O A322364 0,3
%A A322364 _Alois P. Heinz_, Dec 04 2018