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

0,5

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).

Links

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

Formula

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

Example

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;

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

Crossrefs

Cf. A000041, A116645, A116646.

Sequence in context: A238238 A117937 A110897 * A166462 A328177 A320776

Adjacent sequences: A116641 A116642 A116643 * A116645 A116646 A116647

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 20 2006

Status

approved