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

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

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..505

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.

Cf. A000041, A006906, A080130, A177208, A177209, A322380, A322381, A323290, A323291, A323339, A323340.

Sequence in context: A153285 A083557 A119324 * A250034 A379367 A378678

Adjacent sequences: A322361 A322362 A322363 * A322365 A322366 A322367

Keyword

nonn,frac

Author

Alois P. Heinz, Dec 04 2018

Status

approved