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A103919
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Triangle of numbers of partitions of n with total number of odd parts equal to k from {1,..,n}.
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4
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1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 3, 0, 5, 0, 2, 0, 1, 0, 7, 0, 5, 0, 2, 0, 1, 5, 0, 9, 0, 5, 0, 2, 0, 1, 0, 12, 0, 10, 0, 5, 0, 2, 0, 1, 7, 0, 17, 0, 10, 0, 5, 0, 2, 0, 1, 0, 19, 0, 19, 0, 10, 0, 5, 0, 2, 0, 1, 11, 0, 28, 0, 20, 0, 10, 0, 5, 0, 2, 0, 1, 0, 30, 0, 33, 0, 20, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| The partition (0) of n=0 is included. For n>0 no part 0 appears.
The first (k=0) column gives the number of partitions without odd parts, i.e. those with even parts only. See A035363.
Without the alternating zeros this becomes a triangle with columns given by the rows of the S_n(m) table shown in the Riordan reference.
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REFERENCES
| J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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LINKS
| W. Lang: First 11 rows.
D. Kim, A. J. Yee, A note on partitions into distinct parts and odd parts, Ramanujan J. 3 (1999), 227-231. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 11 2008]
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FORMULA
| a(n, k)=number of partitions of n>=0, which have exactly k odd parts (and possibly even parts) for k from {0, ..., n}.
sum(k*T(n,k),k=0..n)=A066897(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
G.f.=G(t,x)=1/product((1-tx^(2j-1))(1-x^(2j)), j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
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EXAMPLE
| [1],[0,1],[1,0,1],[0,2,0,1],[2,0,2,0,1],[0,4,0,2,0,1],...
a(0,0)=1 because n=0 has no odd part, only one even part, 0, by definition. a(5,3)=2 because there are two partitions (1,1,3) and (1,1,1,2) of 5 with exactly 3 odd parts.
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MAPLE
| g:=1/product((1-t*x^(2*j-1))*(1-x^(2*j)), j=1..20): gser:=simplify(series(g, x=0, 22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser, x^n) od: for n from 0 to 18 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
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CROSSREFS
| Row sums A000041 (partition numbers). Columns: k=0: A035363 (with zero entries) A000041 (without zero entries), k=1: A000070, k=2: A000097, k=3: A000098, k=4: A000710.
Cf. A066897.
Sequence in context: A058685 A029300 A096397 * A035445 A053603 A085794
Adjacent sequences: A103916 A103917 A103918 * A103920 A103921 A103922
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 24 2005
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