A263233 Triangle read by rows: T(n,k) is the number of partitions of n having k perfect square parts (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 3, 3, 1, 2, 1, 0, 1, 3, 4, 3, 1, 2, 1, 0, 1, 5, 4, 5, 3, 1, 2, 1, 0, 1, 5, 8, 4, 5, 3, 1, 2, 1, 0, 1, 8, 8, 9, 4, 5, 3, 1, 2, 1, 0, 1, 9, 12, 9, 9, 4, 5, 3, 1, 2, 1, 0, 1, 13, 15, 13, 10, 9, 4, 5, 3, 1, 2, 1, 0, 1

Offset: 0

Author

Emeric Deutsch, Nov 12 2015

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = A000041(n) = number of partitions of n.

T(n,0) = A087153(n).

Sum_{k=0..n}k*T(n,k) = A073336(n) = total number of square parts in all partitions of n.

Examples

			T(8,2) = 5 because we have [6,1,1], [4,4], [4,3,1], [3,3,1,1], [2,2,2,1,1] (the partitions of 8 that have 2 perfect square parts).
Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 2, 1, 0, 1;
  2, 1, 2, 1, 0, 1;
  3, 3, 1, 2, 1, 0, 1;
  3, 4, 3, 1, 2, 1, 0, 1;
  5, 4, 5, 3, 1, 2, 1, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Cf. A000041, A073336, A087153.

Programs

Maple

h:= proc(i) options operator, arrow: i^2 end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form.

Mathematica

Needs["Combinatorica`"]; Table[Count[Replace[#, n_ /; ! IntegerQ@ Sqrt@ n -> Nothing, {1}] & /@ Combinatorica`Partitions@ n, w_ /; Length@ w == k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 19 2015 *)

Formula

G.f.: Product_{i>=1}(1-x^h(i))/((1-x^i)*(1-t*x^h(i))), where h(i) = i^2.