A097304 Triangle of numbers of partitions of n with m parts which are all odd.

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 4, 0, 6, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1

Offset: 1

Author

Wolfdieter Lang, Aug 13 2004

Keywords

• nonn
• tabl
• easy

Examples

			[1];
[0,1];
[1,0,1];
[0,1,0,1];
[1,0,1,0,1];
[0,2,0,1,0,1];
...
T(6,2)=2 because 6 = 1+5 = 3+3; T(6,1) = 0 = T(6,3): there are no partitions of 6 into either one or three parts with only odd numbers;
T(6,4)=1 from 6 = 1+1+1+3; T(6,6)=1 from 6 = 1+1+1+1+1+1.
		

Links

• Álvar Ibeas, First 100 rows, flattened
• W. Lang, First 10 rows.

Crossrefs

Row sums: A000009 (number of partitions of n into odd parts).

Cf. A008284 (partitions of n into k parts).

Cf. A152140.

Programs

Maple

g:=1/product(1-t*x^(2*j-1),j=1..30)-1: gser:=simplify(series(g,x=0,17)): for n from 1 to 15 do P[n]:=sort(coeff(gser,x^n)) od: seq(seq(coeff(P[n],t^j),j=1..n),n=1..15); # Emeric Deutsch, Feb 24 2006

Formula

T(n, m) := 0 if 1 <= n < m, else T(n, m) = number of partitions of n with m parts which are all odd. Hence T(2*k, 2*j-1) = 0, k >= 1, k >= j >= 1; T(2*k-1, 2*j) = 0, k >= 1, k-1 >= j >= 1.

G.f.: 1/Product_{j>=1} (1 - t*x^(2*j-1)). - Emeric Deutsch, Feb 24 2006

T(n, k) = T(n-1, k-1) + T(n-2*k, k). If n+k is even, T(n, k) = A008284((n+k)/2, k) = A072233((n-k)/2, k); 0 otherwise. - Álvar Ibeas, Jul 25 2020