A116687 - OEIS

%I #4 Mar 30 2012 17:36:08

%S 1,2,2,1,3,1,1,2,2,2,1,4,1,3,2,1,2,3,2,4,3,1,4,1,5,3,5,3,1,3,3,2,6,5,

%T 6,4,1,4,2,5,3,9,6,8,4,1,2,3,4,7,5,11,9,9,5,1,6,1,5,5,10,7,15,11,11,5,

%U 1,2,5,2,7,8,13,11,18,15,13,6,1,4,1,9,3,11,10,19,14,24,18,15,6,1,4,3,2,12,5

%N Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts that are smaller than the largest part is equal to k (n>=1, k>=0).

%C Row 1 has one term; row n (n>=2) has n-1 terms. Row sums yield the partition numbers (A000041). T(n,0)=A000005(n) (number of divisors of n). T(n,1)=A032741(n-1) (number of proper divisors of n-1) Sum(k*T(n,k),k=0..n-2)=A116688

%F G.f.=sum(x^i/[(1-x^i)*product(1-t^j*x^j, j=1..i-1), i=1..infinity)].

%e T(6,2)=3 because we have [4,2],[4,1,1] and [2,2,1,1].

%e Triangle starts:

%e 1;

%e 2;

%e 2,1;

%e 3,1,1;

%e 2,2,2,1;

%e 4,1,3,2,1;

%p g:=sum(x^i/(1-x^i)/product(1-(t*x)^j,j=1..i-1),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

%Y Cf. A000041, A000005, A032741, A116688.

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Feb 23 2006