A116644 Triangle read by rows: T(n,k) is the number of partitions of n having exactly k doubletons (n>=0, k>=0). By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).

1, 1, 1, 1, 3, 3, 2, 5, 2, 8, 2, 1, 10, 5, 13, 8, 1, 20, 9, 1, 26, 12, 4, 33, 21, 2, 46, 25, 5, 1, 58, 37, 6, 75, 48, 11, 1, 101, 59, 16, 125, 84, 19, 3, 157, 115, 23, 2, 206, 135, 39, 5, 253, 187, 46, 4, 317, 238, 63, 8, 1, 403, 292, 90, 7, 494, 382, 108, 17, 1, 608, 490, 139, 18

Offset: 0

Author

Emeric Deutsch, Feb 20 2006

Keywords

• nonn
• tabf

Comments

Apparently, rows n with p(p+1)<=n<(p+1)(p+2) have at most p+1 terms. Row sums are the partition numbers (A000041). T(n,0)=A116645(n). Sum(k*T(n,k),k>=0)=A116646(n).

Examples

			T(6,2) = 1 because [2,2,1,1] is the only partition of 6 with 2 doubletons.
Triangle starts:
1;
1;
1,  1;
3;
3,  2;
5,  2;
8,  2, 1;
10, 5;
13, 8, 1;
		

Links

• Alois P. Heinz, Rows n = 0..614, flattened

Crossrefs

Cf. A000041, A116645, A116646.

Programs

Maple

g:=product(1+x^j+t*x^(2*j)+x^(3*j)/(1-x^j),j=1..35): gser:=simplify(series(g,x=0,35)): P[0]:=1: for n from 1 to 24 do P[n]:=coeff(gser,x^n) od: for n from 0 to 24 do seq(coeff(P[n],t,j),j=0..degree(P[n])) od; # sequence given in triangular form

Formula

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