A323339 Numerator of the sum of inverse products of parts in all compositions of n.

1, 1, 3, 7, 11, 347, 3289, 1011, 38371, 136553, 4320019, 12528587, 40771123, 29346499543, 129990006917, 1927874590951, 903657004321, 437445829053473, 12456509813711881, 187206004658210129, 1974369484466728177, 1967745662306280217, 21401375717067880189

Offset: 0

Author

Alois P. Heinz, Jan 11 2019

Keywords

• nonn
• frac

Comments

Numerators of the INVERT transform of reciprocal integers.

Examples

			1/1, 1/1, 3/2, 7/3, 11/3, 347/60, 3289/360, 1011/70, 38371/1680, 136553/3780, 4320019/75600, 12528587/138600, 40771123/285120, ... = A323339/A323340
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..466

Crossrefs

Denominators: A323340.

Cf. A000142, A007840, A011782, A088305, A177208, A177209, A322364, A322365, A322380, A322381, A323290, A323291.

Programs

Maple

b:= proc(n) option remember;
     `if`(n=0, 1, add(b(n-j)/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25);

Mathematica

nmax = 20; Numerator[CoefficientList[Series[1/(1 + Log[1-x]), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Feb 12 2024 *)

Formula

G.f. for fractions: 1 / (1 + log(1 - x)). - Ilya Gutkovskiy, Nov 12 2019

a(n) = numerator( A007840(n)/n! ). - Alois P. Heinz, Jan 04 2024

A323339(n)/A323340(n) ~ exp(n) / (exp(1) - 1)^(n+1). - Vaclav Kotesovec, Feb 12 2024