login
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 difference between the largest part and the smallest part is k (n >= 0, k >= 0).
1

%I #15 Nov 27 2019 02:48:48

%S 1,2,2,1,3,1,2,3,4,2,1,2,5,1,4,4,2,3,6,4,4,6,4,1,2,9,6,1,6,6,9,2,2,11,

%T 10,3,4,10,11,6,4,11,17,6,1,5,11,17,10,1,2,15,21,12,2,6,12,24,18,3,2,

%U 17,28,20,5,6,14,31,26,8,4,17,38,31,10,1,4,18,37,41,14,1,2,21,45,45,19,2,8

%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 occurs and the difference between the largest part and the smallest part is k (n >= 0, k >= 0).

%C Row n contains ceiling((sqrt(9+8n)-3)/2) terms, i.e., 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, ...

%C Row sums yield A034296.

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

%F T(n,0) = A000005(n) (number of divisors of n).

%F Sum_{k>=0} k*T(n,k) = A117471(n).

%e T(9,2)=4 because we have [4,3,2],[3,3,2,1],[3,2,2,1,1] and [3,2,1,1,1,1].

%e Triangle starts:

%e 1;

%e 2;

%e 2, 1;

%e 3, 1;

%e 2, 3;

%e 4, 2, 1;

%e 2, 5, 1;

%p g:=sum(x^j*product(1+t*x^i,i=1..j-1)/(1-x^j),j=1..30): gser:=simplify(series(g,x=0,28)): for n from 1 to 28 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 25 do seq(coeff(P[n],t,j),j=0..ceil((sqrt(9+8*n)-5)/2)) od; # yields sequence in triangular form

%Y Cf. A034296, A000005, A117471.

%K nonn,tabf

%O 1,2

%A _Emeric Deutsch_, Mar 20 2006