A143862 Number of compositions of n such that every part is divisible by number of parts.

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 9, 1, 7, 7, 9, 1, 19, 1, 14, 16, 11, 1, 43, 2, 13, 29, 34, 1, 56, 1, 51, 46, 17, 16, 130, 1, 19, 67, 139, 1, 105, 1, 142, 162, 23, 1, 315, 2, 151, 121, 246, 1, 219, 211, 321, 154, 29, 1, 1021, 1, 31, 219, 488, 496, 495, 1, 594, 232, 834, 1, 1439, 1

Offset: 0

Author

Vladeta Jovovic, Sep 03 2008

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A143773, A261605.

Programs

PARI

{a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d-1,d-1) ))}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018

Formula

G.f.: Sum_{k>=0} x^(k^2) / (1 - x^k)^k.

G.f.: 1 + Sum_{n>=1} (1 + x^n)^(n-1) * x^n. - Paul D. Hanna, Jul 09 2019

a(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018

G.f.: 1 + Sum_{n>=1} (x^n/(1-x^n))^n (conjecture). - Joerg Arndt, Jan 04 2024

For prime p, a(p) = 1, a(2*p) = p and a(p^2) = 2. - Peter Bala, Mar 02 2025

Extensions

More terms from Franklin T. Adams-Watters, Apr 09 2009