A107461 - OEIS

%I #16 Apr 14 2014 06:37:08

%S 1,1,3,1,3,7,3,1,9,25,3,7,3,25,129,1,3,31,3,121,729,25,3,7,123,25,729,

%T 5041,3,151,3,1,729,25,5163,40327,3,25,729,121,3,5071,3,40321,363729,

%U 25,3,7,5043,145,729,40321,3,362911,3628923,5041,729,25,3,40447,3,25

%N Number of gap-free compositions of n into distinct parts, cf. A107428.

%H Alois P. Heinz, <a href="/A107461/b107461.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>0} k!*x^(k*(k+1)/2)/(1-x^k).

%e a(6) = 7 because we have 6, 123, 132, 213, 231, 312 and 321.

%p G:=sum(k!*x^(k*(k+1)/2)/(1-x^k),k=1..20): Gser:=series(G,x=0,73): seq(coeff(Gser,x^n),n=1..70); # _Emeric Deutsch_

%t nn=62;Drop[CoefficientList[Series[Sum[k!x^(k (k+1)/2)/(1-x^k),{k,1,nn}],{x,0,nn}],x],1] (* _Geoffrey Critzer_, Apr 13 2014 *)

%o (PARI)

%o N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);

%o gf=sum(n=1,S, n! * q^(n*(n+1)/2) / (1-q^n) );

%o Vec(gf)

%o /* _Joerg Arndt_, Oct 20 2012 */

%K easy,nonn,look

%O 1,3

%A _Vladeta Jovovic_, May 26 2005

%E More terms from _Emeric Deutsch_, Jun 19 2005