

A116687


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).


1



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, 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, 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
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OFFSET

1,2


COMMENTS

Row 1 has one term; row n (n>=2) has n1 terms. Row sums yield the partition numbers (A000041). T(n,0)=A000005(n) (number of divisors of n). T(n,1)=A032741(n1) (number of proper divisors of n1) Sum(k*T(n,k),k=0..n2)=A116688


LINKS



FORMULA

G.f.=sum(x^i/[(1x^i)*product(1t^j*x^j, j=1..i1), i=1..infinity)].


EXAMPLE

T(6,2)=3 because we have [4,2],[4,1,1] and [2,2,1,1].
Triangle starts:
1;
2;
2,1;
3,1,1;
2,2,2,1;
4,1,3,2,1;


MAPLE

g:=sum(x^i/(1x^i)/product(1(t*x)^j, j=1..i1), 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..n2) od; # yields sequence in triangular form


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



