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+1))).
EXAMPLE
The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
T(5,1) = 2, counting [1,1,1,2] and [2,3].
T(8,2) = 1, counting [2,3,3].
Triangle starts:
1;
1;
1, 1;
2, 1;
4, 1;
5, 2;
7, 4;
...
MAPLE
g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): 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, Oct 10 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, Dec 08 2016 after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Oct 10 2016
STATUS
approved