A117468 - OEIS

%I #34 Aug 29 2016 12:27:14

%S 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,

%T 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,

%U 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

%N 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).

%C 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).

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

%F 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

%e 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].

%e Triangle starts:

%e 1;

%e 1,1;

%e 1,1,1;

%e 1,2,0,1;

%e 1,2,1,0,1;

%e 1,3,2,0,0,1;

%p 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

%p # second Maple program:

%p T:= proc(n, k) option remember; `if`(k<1 or k>n, 0,

%p       `if`(n=k, 1, T(n-k, k) +T(n-k, k-1)))

%p     end:

%p seq(seq(T(n,k), k=1..n), n=1..20);  # _Alois P. Heinz_, Jul 19 2016

%t 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 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 *)

%Y Cf. A034296, A117469.

%K nonn,tabl

%O 1,8

%A _Emeric Deutsch_, Mar 19 2006