Voozh

A322380

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

1, 1, 1, 5, 7, 37, 79, 173, 101, 127, 1033, 1571, 200069, 2564519, 5126711, 25661369, 532393, 431100529, 1855391, 1533985991, 48977868113, 342880481117, 342289639579, 435979161889, 1308720597671, 373092965489, 7824703695283, 24141028973, 31250466692609

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1268

Andreas B. G. Blobel, An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k), arXiv:1904.07808 [math.CO], 2019.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 137.

A. Knopfmacher and 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)/A322381(n) = exp(-gamma) = A080130.

Sum_{n>=0} a(n)/A322381(n)*x^n = Product_{i>=1} (1 + x^i/i). - Geoffrey Critzer, Feb 23 2022

EXAMPLE

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

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

b(n, i-1) +b(n-i, min(i-1, 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, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];

a[n_] := Numerator[b[n, n]];

a /@ Range[0, 30] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

CROSSREFS

Denominators: A322381.