A117468 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 k occurs (1<=k<=n).

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 2, 0, 0, 1, 1, 3, 2, 1, 0, 0, 1, 1, 4, 3, 1, 0, 0, 0, 1, 1, 4, 5, 1, 1, 0, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 0, 0, 1, 1, 5, 7, 3, 0, 1, 0, 0, 0, 0, 1, 1, 6, 9, 4, 1, 1, 0, 0, 0, 0, 0, 1, 1, 6, 10, 6, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 7, 12, 7, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset

1,8

Comments

Also number of partitions of n having k parts and such that all parts smaller than the largest part occur only once. Row sums yield A034296. T(n,1)=T(n,n)=1. Sum_{k=1..n} k*T(n,k) = A117469(n).

Links

Table of n, a(n) for n=1..105.

Formula

G.f.: G(t,x) = Sum_{j>0} t*x^j*(Product_{i=1..j-1} 1+t*x^i)/(1-t*x^j).

T(n,k) = T(n-k,k) + T(n-k,k-1) where T(n,n)=1 and T(n,k)=0 when n<k. - Tricia Muldoon Brown, Jul 19 2016

Example

T(10,3) = 5 because we have [3,3,2,2],[3,3,2,1,1],[3,2,2,2,1],[3,2,2,1,1,1] and [3,2,1,1,1,1,1].
Triangle starts:
1;
1,1;
1,1,1;
1,2,0,1;
1,2,1,0,1;
1,3,2,0,0,1;

Maple

g:=sum(t*x^j*product(1+t*x^i, i=1..j-1)/(1-t*x^j), j=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
# second Maple program:
T:= proc(n, k) option remember; `if`(k<1 or k>n, 0,
      `if`(n=k, 1, T(n-k, k) +T(n-k, k-1)))
    end:
seq(seq(T(n, k), k=1..n), n=1..20);  # Alois P. Heinz, Jul 19 2016

Mathematica

Table[Count[IntegerPartitions@ n, m_ /; And[SubsetQ[m, Range[Min@ m, k]], Max@ m <= k]], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)
T[_, 1] = 1; T[n_, n_] = 1; T[n_, k_] /; 1<k<n := T[n, k] = T[n-k, k] + T[n - k, k-1]; T[_, _] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 29 2016, adapted from Maple *)

Crossrefs

Cf. A034296, A117469.

Sequence in context: A348157 A108455 A193759 * A340453 A116374 A181953

Adjacent sequences: A117465 A117466 A117467 * A117469 A117470 A117471

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 19 2006

Status

approved