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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).
2
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, 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, 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
OFFSET
1,18
COMMENTS
Row sums yield A034296.
Sum_{1..n} k * T(n,k) = A117457(n).
T(2n,n) gives A000009(n). - Alois P. Heinz, Oct 09 2020
LINKS
FORMULA
G.f.: G(t,x) = Sum_{j>=1} t^j*x^j/(1-x^j) * Product_{i=1..j-1} (1+x^i).
EXAMPLE
T(10,5) = 3 because we have [3,3,2,1,1], [3,2,2,2,1] and [2,2,2,2,2].
Triangle starts:
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;
1,1,1,2,2,1,1,1;
...
MAPLE
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
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(x^j*b(n-i*j, i-1), j=1..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(b(n, k), k=0..n)):
seq(T(n), n=1..14); # Alois P. Heinz, Oct 09 2020
MATHEMATICA
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[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][Sum[b[n, k], {k, 0, n}]];
Array[T, 14] // Flatten (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Mar 18 2006
STATUS
approved