A117466 - OEIS

%I #15 Feb 19 2015 05:21:29

%S 1,1,1,2,0,1,2,1,0,1,3,1,0,0,1,4,1,1,0,0,1,5,1,1,0,0,0,1,6,2,0,1,0,0,

%T 0,1,8,2,1,1,0,0,0,0,1,10,2,1,0,1,0,0,0,0,1,12,3,1,0,1,0,0,0,0,0,1,15,

%U 3,2,1,0,1,0,0,0,0,0,1,18,4,1,1,0,1,0,0,0,0,0,0,1,22,5,1,1,0,0,1,0,0,0,0,0

%N Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part k to the largest part occurs (1<=k<=n).

%C Row sums yield A034296. T(n,1) = A000009(n). sum(k*T(n,k), k=1..n) = A117467(n).

%H Alois P. Heinz, <a href="/A117466/b117466.txt">Rows n = 1..141, flattened</a>

%F G.f.: G(t,x) = sum(tx^j*product(1+x^i, i=1..j-1)/(1-tx^j), j >=1).

%e T(11,2) = 3 because we have [4,3,2,2], [3,3,3,2] and [3,2,2,2,2].

%e Triangle starts:

%e 1;

%e 1,1;

%e 2,0,1;

%e 2,1,0,1;

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

%e 4,1,1,0,0,1;

%p g:= sum(t*x^j*product(1+x^i,i=1..j-1)/(1-t*x^j),j=1..50): gser:=simplify(series(g,x=0,17)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, k, i) option remember;

%p       `if`(n<0, 0, `if`(n=0, 1, `if`(i<k, 0, b(n, k, i-1)+

%p       `if`(i>n, 0, b(n-i, k, i)) )))

%p     end:

%p T:= (n, k)-> add(b(n-(i+k)*(i+1-k)/2, k, i), i=k..n):

%p seq(seq(T(n, k), k=1..n), n=1..14);  # _Alois P. Heinz_, Jul 06 2012

%t b[n_, k_, i_] := b[n, k, i] = If[n<0, 0, If[n == 0, 1, If[i<k, 0, b[n, k, i-1] + If[i>n, 0, b[n-i, k, i]]]]]; T[n_, k_] := Sum[b[n-(i+k)*(i+1-k)/2, k, i], {i, k, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Feb 19 2015, after _Alois P. Heinz_ *)

%Y Cf. A034296, A000009, A117467.

%K nonn,tabl,look

%O 1,4

%A _Emeric Deutsch_, Mar 19 2006