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

Offset: 0

Author

Alois P. Heinz, Dec 05 2018

Keywords

• nonn
• frac

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

Examples

			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
		

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.

Crossrefs

Denominators: A322381.

Cf. A000009, A000142, A007838, A022629, A080130, A177208, A177209, A322364, A322365, A323290, A323291, A323339, A323340.

Programs

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 *)

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