%I #12 Nov 23 2020 08:03:53
%S 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,
%T 1,1,1,1,2,2,2,2,1,1,1,1,1,1,2,3,2,2,1,1,1,1,1,1,2,3,3,2,2,1,1,1,1,1,
%U 2,2,3,4,3,2,2,1,1,1,1,1,1,2,3,4,4,3,2,2,1,1,1,1,1,1,2,3,4,5,4,3,2,2,1,1,1
%N 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 Row sums yield A034296.
%C Sum_{1..n} k * T(n,k) = A117457(n).
%C T(2n,n) gives A000009(n). - _Alois P. Heinz_, Oct 09 2020
%H Alois P. Heinz, <a href="/A117456/b117456.txt">Rows n = 1..200, flattened</a>
%F G.f.: G(t,x) = Sum_{j>=1} t^j*x^j/(1-x^j) * Product_{i=1..j-1} (1+x^i).
%e T(10,5) = 3 because we have [3,3,2,1,1], [3,2,2,2,1] and [2,2,2,2,2].
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,1,1;
%e 1,1,1,1;
%e 1,1,1,1,1;
%e 1,1,2,1,1,1;
%e 1,1,1,2,1,1,1;
%e 1,1,1,2,2,1,1,1;
%e ...
%p 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
%p # second Maple program:
%p b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p `if`(i<1, 0, add(x^j*b(n-i*j, i-1), j=1..n/i))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(b(n, k), k=0..n)):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Oct 09 2020
%t b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[x^j b[n - i j, i - 1], {j, 1, n/i}]]]];
%t T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][Sum[b[n, k], {k, 0, n}]];
%t Array[T, 14] // Flatten (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y Cf. A000009, A034296, A117457.
%K nonn,tabl
%O 1,18
%A _Emeric Deutsch_, Mar 18 2006