%I A117456
%S A117456 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,
%T A117456 1,1,1,2,2,2,2,1,1,1,1,1,1,2,3,2,2,1,1,1,1,1,1,2,3,3,2,2,1,1,1,1,1,2,2,
%U A117456 3,4,3,2,2,1,1,1,1,1,1,2,3,4,4,3,2,2,1,1,1,1,1,1,2,3,4,5,4,3,2,2,1,1,1
%N A117456 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 number of parts is k (1<=k<=n).
%C A117456 Row sums yield A034296. sum(k*T(n,k),k=1..n)=A117457(n).
%F A117456 G.f.=G(t,x)=sum(t^j*x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..infinity).
%e A117456 T(10,5)=3 because we have [3,3,2,1,1],[3,2,2,2,1] and [2,2,2,2,2].
%e A117456 Triangle starts:
%e A117456 1;
%e A117456 1,1;
%e A117456 1,1,1;
%e A117456 1,1,1,1;
%e A117456 1,1,1,1,1;
%e A117456 1,1,2,1,1,1;
%e A117456 1,1,1,2,1,1,1;
%e A117456 1,1,1,2,2,1,1,1;
%p A117456 g:=sum(t^j*x^j*product(1+x^i,i=1..j-1)/(1-x^j),j=1..60): gser:=simplify(series(g,
               x=0,55)): for n from 1 to 15 do P[n]:=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
%Y A117456 Cf. A034296, A117457.
%Y A117456 Sequence in context: A030379 A030392 A165633 this_sequence A030621 A120336 
               A039738
%Y A117456 Adjacent sequences: A117453 A117454 A117455 this_sequence A117457 A117458 
               A117459
%K A117456 nonn,tabl
%O A117456 1,18
%A A117456 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 18 2006