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

1,17

Links

Álvar Ibeas, First 100 rows, flattened

W. Lang, First 10 rows.

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

Example

[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.

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

Crossrefs

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

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

Cf. A152140.

Sequence in context: A287179 A236511 A235924 * A136745 A214157 A246720

Adjacent sequences: A097301 A097302 A097303 * A097305 A097306 A097307

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Aug 13 2004

Status

approved