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Search: id:A097304
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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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2
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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
(list; table; graph; listen)
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OFFSET
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1,17
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COMMENT
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Row sums: A000009 (number of partitions of n into odd parts).
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LINKS
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W. Lang, First 10 rows.
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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(1-tx^(2j-1),j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2006
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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 (deutsch(AT)duke.poly.edu), Feb 24 2006
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CROSSREFS
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Cf. A008284 (partitions of n into k parts).
Sequence in context: A033764 A033784 A082886 this_sequence A136745 A090465 A052344
Adjacent sequences: A097301 A097302 A097303 this_sequence A097305 A097306 A097307
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 13 2004
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