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Search: id:A116856
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| A116856 |
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Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1). |
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+0
3
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| 1, 1, 1, 0, 1, 2, 2, 0, 0, 0, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 1, 5, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 8, 0, 1, 0, 1, 10, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 12, 0, 2, 0, 1, 15, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 0, 2, 0, 1, 0, 1, 22, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 0, 3, 0, 1, 0, 1, 32, 0, 4, 0, 1, 0
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Row 2n-1 has 2n-1 terms; row 4n+2 has 2n+1 terms; row 4n has 2n-1 terms. Row sums yield A000009. T(n,2k)=0. Sum(k*T(n,k),k>=1)=A092314(n)
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FORMULA
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G.f.=sum(t^(2j-1)*x^(2j-1)/product(1-x^(2i-1), i=j..infinity), j=1..infinity).
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EXAMPLE
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T(12,3)=2 because we have [9,3] and [3,3,3,3].
Triangle starts:
1;
1;
1,0,1;
2;
2,0,0,0,1;
3,0,1;
4,0,0,0,0,0,1
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MAPLE
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g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1), i=j..20), j=1..30): gser:=simplify(series(g, x=0, 20)): for n from 1 to 17 do P[n]:=sort(coeff(gser, x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n mod 4 = 2 then n/2 else n/2-1 fi end: for n from 1 to 17 do seq(coeff(P[n], t^j), j=1..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]
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CROSSREFS
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Cf. A000009, A092314, A116799.
Sequence in context: A049800 A131018 A035395 this_sequence A140344 A024158 A054978
Adjacent sequences: A116853 A116854 A116855 this_sequence A116857 A116858 A116859
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 24 2006
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