A116858 Triangle read by rows: T(n,k) is the number of partitions into distinct parts having Durfee square of size k (n>=1, k>=1).

1, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 4, 2, 6, 2, 8, 2, 10, 2, 12, 1, 2, 14, 2, 2, 16, 4, 2, 18, 7, 2, 20, 10, 2, 22, 14, 2, 24, 20, 2, 26, 26, 2, 28, 34, 2, 30, 44, 2, 32, 54, 1, 2, 34, 66, 2, 2, 36, 80, 4, 2, 38, 94, 8, 2, 40, 110, 13, 2, 42, 128, 20, 2, 44, 146, 30, 2, 46, 166, 42, 2, 48, 188

Offset

1,3

Comments

Row n has floor([1+sqrt(1+24n)]/6) terms. Row sums yield A000009. Sum(k*T(n,k),k>=0)=A116859(n).

Links

Table of n, a(n) for n=1..83.

Formula

G.f.=sum(t^k*x^(k(3k-1)/2)*(1+x^(2k))*product((1+x^j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity).

Example

T(8,2)=4 because we have [6,2], [5,3], [5,2,1] and [4,3,1].
Triangle starts:
1;
1;
2;
2;
2,1;
2,2;
2,3;
2,4;

Maple

g:=sum(t^k*x^(k*(3*k-1)/2)*(1+x^(2*k))*product((1+x^j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..10): gser:=simplify(series(g, x=0, 36)): for n from 1 to 32 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 32 do seq(coeff(P[n], t^j), j=1..floor((1+sqrt(1+24*n))/6)) od; # yields sequence in triangular form

Crossrefs

Cf. A000009, A116859.

Sequence in context: A274138 A179301 A008334 * A182134 A189684 A308176

Adjacent sequences: A116855 A116856 A116857 * A116859 A116860 A116861

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 26 2006

Status

approved