A113685 Triangular array read by rows: T(n,k) is the number of partitions of n in which sum of odd parts is k, for k=0,1,...,n; n>=0.

1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 4, 0, 3, 0, 4, 0, 3, 0, 5, 5, 0, 3, 0, 4, 0, 4, 0, 6, 0, 5, 0, 6, 0, 6, 0, 5, 0, 8, 7, 0, 5, 0, 6, 0, 8, 0, 6, 0, 10, 0, 7, 0, 10, 0, 9, 0, 10, 0, 8, 0, 12, 11, 0, 7, 0, 10, 0, 12, 0, 12, 0, 10, 0, 15, 0, 11, 0, 14, 0, 15, 0

Offset: 0

Author

Clark Kimberling, Nov 05 2005

Keywords

• nonn
• tabl

Comments

(Sum over row n) = A000041(n) = number of partitions of n.

Reversal of this array is array in A113686.

From Gary W. Adamson, Apr 11 2010: (Start)

Let M = an infinite lower triangular matrix with A000041 interleaved with zeros: (1, 0, 1, 0, 2, 0, 3, 0, 5, ...) and Q = A000009 diagonalized with the rest zeros.

Then A113685 = M*Q. That row sums of the triangle (deleting prefaced zeros) = A000041 is equivalent to the identity: p(x) = p(x^2) * A000009(x). (End)

Examples

			First 5 rows:
  1;
  0, 1;
  1, 0, 1;
  0, 1, 0, 2;
  2, 0, 1, 0, 2;
  0, 2, 0, 2, 0, 3.
The partitions of 5 are 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1.
The sums of odd parts are 5,1,3,5,1,3,5, respectively, so that the numbers of 0's, 1's, 2s, 3s, 4s, 5s are 0,2,0,2,0,3, which is row 5 of the array.
		

Crossrefs

Cf. A000041, A113686, A066967.

Programs

Maple

g := 1/product((1-t^(2*j-1)*x^(2*j-1))*(1-x^(2*j)),j=1..20):
gser := simplify(series(g,x=0,22)):
P[0] := 1: for n from 1 to 14 do P[n] := coeff(gser,x^n) od:
for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..n) od;
# yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

Formula

G.f.: G(t,x) = 1/Product_{j>=1} (1 - t^(2j-1)*x^(2j-1))*(1-x^(2j)). - Emeric Deutsch, Feb 17 2006

Extensions

More terms from Emeric Deutsch, Feb 17 2006