A026821 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the least being k, for k=1,2,...,n.

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 0, 1, 5, 2, 1, 1, 0, 0, 0, 0, 0, 1, 5, 3, 1, 1, 1, 0, 0, 0, 0, 0, 1, 7, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 8, 4, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

T(n,1) = A025147(n-1). Sum(k*T(n,k),k=1..n) = A092265(n). - Emeric Deutsch, Feb 24 2006

Examples

			T(11,2)=3 because we have [9,2],[6,3,2] and [5,4,2].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,1;
1,1,0,0,1;
		

Crossrefs

Cf. A025147, A092265.

Programs

Maple

g:=sum(t^j*x^j*product(1+x^i,i=j+1..50),j=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: seq(seq(coeff(P[n],t^j),j=1..n),n=1..14); # Emeric Deutsch, Feb 24 2006

Formula

T(n, k) = T(n-k, k+1) + ... + T(n-k, n-k) for 1<=k<=m and T(n, k)=0 for m+1<=k<=n-1, where m=[ (n-1)/2 ]; T(n, n)=1 for n >= 1.

G.f.: sum(t^j*x^j*product(1+x^i,i=j+1..infinity),j=1..infinity). - Emeric Deutsch, Feb 24 2006