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

0,10

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)

Links

Table of n, a(n) for n=0..97.

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

Example

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.

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

Crossrefs

Cf. A000041, A113686, A066967.

Sequence in context: A366475 A259976 A377415 * A049825 A287443 A039651

Adjacent sequences: A113682 A113683 A113684 * A113686 A113687 A113688

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 05 2005

Extensions

More terms from Emeric Deutsch, Feb 17 2006

Status

approved