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
KEYWORD
AUTHOR
Wolfdieter Lang, Aug 13 2004
STATUS
approved