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A322364 Numerator of the sum of inverse products of parts in all partitions of n. 8
1, 1, 3, 11, 7, 27, 581, 4583, 2327, 69761, 775643, 147941, 30601201, 30679433, 10928023, 6516099439, 445868889691, 298288331489, 7327135996801, 1029216937671847, 14361631943741, 837902013393451, 2766939485246012129, 274082602410356881, 835547516381094139939 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
A. Knopfmacher, J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
D. Zeilberger, N. Zeilberger, Fractional Counting of Integer Partitions, 2018.
FORMULA
Limit_{n-> infinity} a(n)/(n*A322365(n)) = exp(-gamma) = A080130.
EXAMPLE
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
MAPLE
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1) +b(n-i, min(i, n-i))/i)
end:
a:= n-> numer(b(n$2)):
seq(a(n), n=0..30);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0||i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]/i];
a[n_] := Numerator[b[n, n]];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)
PROG
(PARI) a(n) = {my(s=0); forpart(p=n, s += 1/vecprod(Vec(p))); numerator(s); } \\ Michel Marcus, Apr 29 2020
CROSSREFS
Denominators: A322365.
Sequence in context: A153285 A083557 A119324 * A250034 A006495 A112286
KEYWORD
nonn,frac
AUTHOR
Alois P. Heinz, Dec 04 2018
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)