A026836 Triangular array T read by rows: T(n,k) = number of partitions of n into distinct parts, the greatest being k, for k=1,2,...,n.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

Conjecture: A199918(n) = Sum_{k=1..n} (-1)^(n-k) T(n,k). - George Beck, Jan 13 2019

Examples

			Triangle begins:
[1]
[0, 1]
[0, 1, 1]
[0, 0, 1, 1]
[0, 0, 1, 1, 1]
[0, 0, 1, 1, 1, 1]
[0, 0, 0, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 1, 1, 1]
[0, 0, 0, 1, 2, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 2, 3, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 3, 2, 2, 1, 1, 1]
[0, 0, 0, 0, 1, 3, 4, 4, 3, 2, 2, 1, 1, 1]
... - _N. J. A. Sloane_, Nov 09 2018
		

Links

• Henry Bottomley, Partition calculators using java applets
• Index entries for sequences related to partitions

Crossrefs

If seen as a square array then transpose of A070936 and visible form of A053632. Central diagonal and those to the right of center are A000009 as are row sums.

Programs

Maple

with(combinat);
f2:=proc(n) local i,j,p,t0,t1,t2;
t0:=Array(1..n,0);
t1:=partition(n);
p:=numbpart(n);
for i from 1 to p do
t2:=t1[i];
if nops(convert(t2,set))=nops(t2) then
# now have a partition t2 of n into distinct parts
t0[t2[-1]]:=t0[t2[-1]]+1;
od:
[seq(t0[j],j=1..n)];
end proc;
for n from 1 to 12 do lprint(f2(n)); od: # N. J. A. Sloane, Nov 09 2018

Formula

T(n, k) = A070936(n-k, k-1) = A053632(k-1, n-k) = T(n-1, k-1)+T(n-2k+1, k-1). - Henry Bottomley, May 12 2002

T(n, k) = coefficient of x^n in x^k*Product_{i=1..k-1} (1+x^i). - Vladeta Jovovic, Aug 07 2003