|
|
A268190
|
|
Triangle read by rows: T(n,k) (n, k>=1) is the number of partitions of n such that the difference between the two largest distinct parts is k; T(n,0) is the number of partitions of n in which all parts are equal.
|
|
2
|
|
|
1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 3, 2, 1, 1, 2, 6, 3, 2, 1, 1, 4, 7, 5, 2, 2, 1, 1, 3, 11, 5, 5, 2, 2, 1, 1, 4, 13, 10, 5, 4, 2, 2, 1, 1, 2, 20, 11, 8, 5, 4, 2, 2, 1, 1, 6, 23, 16, 10, 8, 4, 4, 2, 2, 1, 1, 2, 33, 20, 15, 9, 8, 4, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A000005(n) = number of divisors of n.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: G(t,x) = Sum_{k>0} (x^k/(1-x^k)) + Sum_{k>1} (Sum_{j=1..i-1} t^{i-j}*x^{i+j}/((1 - x^i)*Product_{k=1..j} (1 - x^k))).
|
|
EXAMPLE
|
T(5,0)=2 because we have [5] and [1,1,1,1,1]; T(5,1)=3 because we have [3,2], [2,2,1], and [2,1,1,1]; T(5,2)=1 because we have [3,1,1]; T(5,3)=1 because we have [4,1].
Triangle starts:
1;
2;
2,1;
3,1,1;
2,3,1,1;
4,3,2,1,1;
|
|
MAPLE
|
G := add(x^k/(1-x^k), k = 1 .. 80)+ add(add(t^(i-j)*x^(i+j)/((1-x^i)*mul(1-x^k, k = 1 .. j)), j = 1 .. i-1), i = 2 .. 80): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: 1; for n from 2 to 25 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, l, i) option remember; `if`(irem(n, i)=0, x^
`if`(l=0, 0, i-l), 0) +`if`(i>n, 0, add(b(n-i*j,
`if`(j=0, l, i), i+1), j=0..(n-1)/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
|
|
MATHEMATICA
|
b[n_, l_, i_] := b[n, l, i] = If[Mod[n, i] == 0, x^If[l == 0, 0, i-l], 0] + If[i>n, 0, Sum[b[n-i*j, If[j == 0, l, i], i+1], {j, 0, (n-1)/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|