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

Author

Emeric Deutsch, Feb 23 2006

Keywords

• nonn
• tabf

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=0..n-2} k*T(n,k) = A116688(n).

Examples

			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;
  ...
		

Crossrefs

Cf. A000041, A000005, A032741, A116688.

Programs

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

Formula

G.f.: Sum_{i>=1} x^i/((1-x^i)*Product_{j=1..i-1} (1-t^j*x^j)).