A171628 Number of compositions of n such that the smallest part is divisible by the number of parts.

1, 1, 1, 2, 3, 3, 3, 4, 6, 8, 11, 15, 19, 22, 25, 30, 37, 47, 62, 83, 108, 136, 168, 205, 247, 295, 354, 429, 524, 642, 789, 972, 1196, 1466, 1789, 2173, 2625, 3155, 3778, 4515, 5391, 6437, 7692, 9201, 11014, 13186, 15780, 18865, 22516, 26818, 31871, 37791

Offset: 1

Author

Vladeta Jovovic, Dec 13 2009

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..900

Crossrefs

Cf. A168656, A171625.

Programs

Maple

b:= proc(n,t,g) option remember; `if` (n=0, `if` (irem(g, t)=0, 1, 0), add (b(n-i, t+1, min(i, g)), i=1..n)) end: a:= n-> b(n,0,infinity): seq (a(n), n=1..60); # Alois P. Heinz, Dec 15 2009
A171628 := proc(n) local g,k; g := 0 ; for k from 0 to n do g := g+add (x^(k*d)*(1-x^d)/(1-x)^d,d=numtheory[divisors](k)) ; g := expand(g) ; end do ; coeftayl(g,x=0,n) ; end proc: seq(A171628(n),n=1..60) ; # R. J. Mathar, Dec 14 2009

Mathematica

b[n_, t_, g_] := b[n, t, g] = If[n == 0, If [Mod[g, t] == 0, 1, 0], Sum[b[n - i, t + 1, Min[i, g]], {i, n}]];
a[n_] := b[n, 0, Infinity];
Array[a, 60] (* Jean-François Alcover, May 23 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{n>=0} [Sum_{d|n} x^(n*d)*(1-x^d)/(1-x)^d].

Extensions

More terms from R. J. Mathar and Alois P. Heinz, Dec 14 2009