OFFSET
0,5
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..1000, flattened
FORMULA
G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).
EXAMPLE
The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
T(5,1) = 1, counting [1,1,3].
T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
T(8,2) = 1, counting [1,1,2,2,2].
Triangle starts:
1;
1;
1,1;
3;
4,1;
6,1;
8,3.
MAPLE
G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
`if`(n=0, 1, `if`(i<1, 0, add(
`if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30); # Alois P. Heinz, Sep 30 2016
MATHEMATICA
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 30 2016
STATUS
approved