A118806 Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0).

1, 1, 2, 2, 1, 5, 6, 1, 9, 2, 12, 3, 19, 3, 24, 5, 1, 34, 8, 43, 13, 62, 13, 2, 77, 23, 1, 105, 28, 2, 132, 40, 4, 177, 49, 5, 220, 71, 6, 287, 89, 8, 1, 356, 123, 11, 462, 147, 18, 570, 198, 23, 1, 723, 249, 29, 1, 888, 329, 37, 1, 1121, 400, 50, 4, 1370, 518, 69, 1, 1705, 642, 85

Offset: 0

Author

Emeric Deutsch, Apr 29 2006

Keywords

• nonn
• tabf

Comments

T(n,0)=A118807(n). T(n,1)=A118808(n). Row sums yield the partition numbers (A000041). Sum(k*T(n,k), k>=0)=A117524(n) (n>=1).

Examples

			T(12,2) = 2 because we have [3,3,3,1,1,1] and [3,2,2,2,1,1,1].
Triangle starts:
1;
1;
2;
2,  1;
5;
6,  1;
9,  2;
12, 3;
		

Links

• Alois P. Heinz, Rows n = 0..703

Crossrefs

Cf. A000041, A118807, A118808, A117524, A116595, A116644.

Programs

Maple

g:=product(1+x^j+x^(2*j)+t*x^(3*j)+x^(4*j)/(1-x^j),j=1..35): gser:=simplify(series(g,x=0,35)): P[0]:=1: for n from 1 to 30 do P[n]:=coeff(gser,x^n) od: for n from 0 to 30 do seq(coeff(P[n],t,j),j=0..degree(P[n])) od; # sequence given in triangular form

Formula

G.f.: product(1+x^j+x^(2j)+tx^(3j)+x^(4j)/(1-x^j), j=1..infinity).