A116663 Triangle read by rows: T(n,k) = number of partitions of n into odd parts and having exactly k parts equal to 1 (n>=0, 1<=k<=n).

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1

Offset

0,46

Comments

Row sums yield A000009. T(n,0)=A087897(n). Column k has g.f.=x^k/Product(1-x^(2j-1), j=2..infinity) (all columns are basically identical). Sum(k*T(n,k),k=0..n)=A036469(n).

Links

Table of n, a(n) for n=0..104.

Formula

G.f.=1/[(1-tx)*Product(1-x^(2j-1), j=2..infinity)].

Example

T(10,1)=2 because the only partitions of 10 into odd parts and having exactly 1 part equal to 1 are [9,1] and [3,3,3,1].
Triangle starts:
1;
0,1;
0,0,1;
1,0,0,1;
0,1,0,0,1;

Maple

g:=1/(1-t*x)/product(1-x^(2*j-1), j=2..30): gser:=simplify(series(g, x=0, 18)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(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

Crossrefs

Cf. A000009, A087897, A036469.

Sequence in context: A251926 A335504 A037908 * A258940 A340607 A319659

Adjacent sequences: A116660 A116661 A116662 * A116664 A116665 A116666

Keyword

nonn,tabl

Author

Emeric Deutsch, Feb 22 2006

Status

approved