A117466 Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part k to the largest part occurs (1<=k<=n).

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 5, 1, 1, 0, 0, 0, 1, 6, 2, 0, 1, 0, 0, 0, 1, 8, 2, 1, 1, 0, 0, 0, 0, 1, 10, 2, 1, 0, 1, 0, 0, 0, 0, 1, 12, 3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 15, 3, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 18, 4, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 22, 5, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Author

Emeric Deutsch, Mar 19 2006

Keywords

• nonn
• tabl
• look

Comments

Row sums yield A034296. T(n,1) = A000009(n). sum(k*T(n,k), k=1..n) = A117467(n).

Examples

			T(11,2) = 3 because we have [4,3,2,2], [3,3,3,2] and [3,2,2,2,2].
Triangle starts:
1;
1,1;
2,0,1;
2,1,0,1;
3,1,0,0,1;
4,1,1,0,0,1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A034296, A000009, A117467.

Programs

Maple

g:= sum(t*x^j*product(1+x^i,i=1..j-1)/(1-t*x^j),j=1..50): gser:=simplify(series(g,x=0,17)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, k, i) option remember;
      `if`(n<0, 0, `if`(n=0, 1, `if`(in, 0, b(n-i, k, i)) )))
    end:
T:= (n, k)-> add(b(n-(i+k)*(i+1-k)/2, k, i), i=k..n):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Jul 06 2012

Mathematica

b[n_, k_, i_] := b[n, k, i] = If[n<0, 0, If[n == 0, 1, If[in, 0, b[n-i, k, i]]]]]; T[n_, k_] := Sum[b[n-(i+k)*(i+1-k)/2, k, i], {i, k, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Formula

G.f.: G(t,x) = sum(tx^j*product(1+x^i, i=1..j-1)/(1-tx^j), j >=1).