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A097304
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Triangle of numbers of partitions of n with m parts which are all odd.
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4
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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
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OFFSET
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1,17
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LINKS
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FORMULA
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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
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EXAMPLE
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[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.
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MAPLE
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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
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CROSSREFS
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Row sums: A000009 (number of partitions of n into odd parts).
Cf. A008284 (partitions of n into k parts).
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KEYWORD
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AUTHOR
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STATUS
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approved
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