A116927 Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having k 1's (n>=1,k>=0).

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

Offset

1,60

Comments

Row 1 has 2 terms; row 2 has one term; row 2n-1 has n terms; row 2n has n-1 terms. Row sums yield A000700. Column 0, except for the first term, yields A090723. Sum(k*T(n,k),k>=0)=A116928(n).

Links

Table of n, a(n) for n=1..105.

Formula

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

Example

T(22,3)=2 because we have [8,5,2,2,2,1,1,1] and [7,4,4,4,1,1,1].
Triangle starts:
0,1;
0;
0,1;
1;
0,0,1;
0,1;
0,0,0,1;
1,0,1;

Maple

g:=t*x-x+sum(x^(k^2)/(1-t*x^2)/product(1-x^(2*j), j=2..k), k=1..30): gser:=simplify(series(g, x=0, 35)): for n from 1 to 30 do P[n]:=sort(coeff(gser, x^n)) od: d:=proc(n) if n=1 then 1 elif n=2 then 0 elif n mod 2 = 1 then (n-1)/2 else (n-4)/2 fi end: for n from 1 to 30 do seq(coeff(P[n], t, j), j=0..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]

Crossrefs

Cf. A000700, A090723, A116928.

Sequence in context: A360079 A127523 A364389 * A137276 A287234 A309938

Adjacent sequences: A116924 A116925 A116926 * A116928 A116929 A116930

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 26 2006

Status

approved