A116675 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).

1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 5, 0, 2, 0, 4, 0, 7, 0, 1, 3, 0, 7, 0, 0, 10, 0, 2, 4, 0, 11, 0, 0, 14, 0, 4, 5, 0, 17, 0, 0, 19, 0, 8, 6, 0, 25, 0, 1, 0, 25, 0, 13, 0, 8, 0, 36, 0, 2, 0, 33, 0, 21, 0, 10, 0, 50, 0, 4, 0, 43, 0, 33, 0, 12, 0, 69, 0, 8, 0, 55, 0, 49, 0, 15, 0, 93, 0, 14

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0) = A035457(n) (n>=1); T(2n,0) = A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n-1,1) = A036469(n). Sum(k*T(n,k), k>=0) = A116676(n).

Examples

			T(8,2) = 4 because we have [7,1], [5,3], [5,2,1] and [4,3,1] ([8] and [6,2] do not qualify).
Triangle starts:
1;
0, 1;
1, 0;
0, 2;
1, 0, 1;
0, 3, 0;
		

Links

• Alois P. Heinz, Rows n = 0..750, flattened

Crossrefs

Cf. A000009, A035457, A036469, A116676.

Programs

Maple

g:=product((1+t*x^(2*j-1))*(1+x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 26 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 26 do seq(coeff(P[n],t,j),j=0..floor(sqrt(n))) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)->
      x+y, b(n, i-1), `if`(i>n, [], [`if`(irem(i,2)=0, [][], 0),
      b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= b(n, n); l[], 0$(1+floor(sqrt(n))-nops(l)) end:
seq (T(n), n=0..30);  # Alois P. Heinz, Nov 21 2012

Mathematica

rows = 25; coes = CoefficientList[Product[(1+t*x^(2j-1))(1+x^(2j)), {j, 1, rows}], {x, t}][[1 ;; rows]]; MapIndexed[Take[#1, Floor[Sqrt[#2[[1]]-1]]+1]&, coes] // Flatten (* Jean-François Alcover, May 13 2015 *)

Formula

G.f.: product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).