%I A117454
%S A117454 1,1,1,1,1,0,1,1,1,0,1,1,0,2,0,1,1,1,0,2,0,1,1,0,1,1,2,0,1,1,1,1,1,1,2,
%T A117454 0,1,1,0,1,2,2,1,2,0,1,1,1,0,2,2,2,1,2,0,1,1,0,2,0,3,3,2,1,2,0,1,1,1,0,
%U A117454 2,2,3,3,2,1,2,0,1,1,0,1,2,2,3,4,3,2,1,2,0,1,1,1,1,1,3,4,3,4,3,2,1,2,0
%N A117454 Triangle read by rows: T(n,k) is the number of partitions of n into distinct 
               parts such that the difference between the largest and smallest parts 
               is k (n>=1; 0<=k<=n-2 for n>=2).
%C A117454 Also number of partitions of n in which all integers smaller than the 
               largest part occur and have k parts smaller than the largest part 
               (n>=1, k>=0). Row 1 has one term; rows j (j>=2) have j-1 terms. Row 
               sums yield A000009. sum(k*T(n,k),k=0..n-2)=A117455(n).
%F A117454 G.f.=G(t,x)=sum(t^(i-1)*x^(i(i+1)/2)/[(1-x^i)product(1-tx^j, j=1..i-1)], 
               i=1..infinity).
%e A117454 T(12,5)=3 because we have [7,3,2],[6,5,1] and [6,3,2,1].
%e A117454 Triangle starts:
%e A117454 1;
%e A117454 1;
%e A117454 1,1;
%e A117454 1,0,1;
%e A117454 1,1,0,1;
%e A117454 1,0,2,0,1;
%p A117454 g:=sum(t^(i-1)*x^(i*(i+1)/2)/(1-x^i)/product(1-t*x^j,j=1..i-1),i=1..20): 
               gser:=simplify(series(g,x=0,20)): for n from 1 to 16 do P[n]:=coeff(gser,
               x^n) od: 1; for n from 2 to 16 do seq(coeff(P[n],t,j),j=0..n-2) od; 
               # yields sequence in triangular form
%Y A117454 Cf. A000009, A117455.
%Y A117454 Sequence in context: A024363 A050600 A129691 this_sequence A115357 A171182 
               A063962
%Y A117454 Adjacent sequences: A117451 A117452 A117453 this_sequence A117455 A117456 
               A117457
%K A117454 nonn,tabf
%O A117454 1,14
%A A117454 Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 18 2006