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, 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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..97.

Formula

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

Crossrefs

Cf. A000041, A000005, A032741, A116688.

Sequence in context: A230260 A193262 A120967 * A264033 A236293 A056044

Adjacent sequences: A116684 A116685 A116686 * A116688 A116689 A116690

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 23 2006

Status

approved