A117470 Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part to the largest part occurs and the difference between the largest part and the smallest part is k (n >= 0, k >= 0).

1, 2, 2, 1, 3, 1, 2, 3, 4, 2, 1, 2, 5, 1, 4, 4, 2, 3, 6, 4, 4, 6, 4, 1, 2, 9, 6, 1, 6, 6, 9, 2, 2, 11, 10, 3, 4, 10, 11, 6, 4, 11, 17, 6, 1, 5, 11, 17, 10, 1, 2, 15, 21, 12, 2, 6, 12, 24, 18, 3, 2, 17, 28, 20, 5, 6, 14, 31, 26, 8, 4, 17, 38, 31, 10, 1, 4, 18, 37, 41, 14, 1, 2, 21, 45, 45, 19, 2, 8

Offset

1,2

Comments

Row n contains ceiling((sqrt(9+8n)-3)/2) terms, i.e., 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ...

Row sums yield A034296.

Links

Table of n, a(n) for n=1..89.

Formula

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

T(n,0) = A000005(n) (number of divisors of n).

Sum_{k>=0} k*T(n,k) = A117471(n).

Example

T(9,2)=4 because we have [4,3,2],[3,3,2,1],[3,2,2,1,1] and [3,2,1,1,1,1].
Triangle starts:
  1;
  2;
  2, 1;
  3, 1;
  2, 3;
  4, 2, 1;
  2, 5, 1;

Maple

g:=sum(x^j*product(1+t*x^i, i=1..j-1)/(1-x^j), j=1..30): gser:=simplify(series(g, x=0, 28)): for n from 1 to 28 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 25 do seq(coeff(P[n], t, j), j=0..ceil((sqrt(9+8*n)-5)/2)) od; # yields sequence in triangular form

Crossrefs

Cf. A034296, A000005, A117471.

Sequence in context: A227083 A327571 A166363 * A070786 A255542 A117500

Adjacent sequences: A117467 A117468 A117469 * A117471 A117472 A117473

Keyword

nonn,tabf

Author

Emeric Deutsch, Mar 20 2006

Status

approved