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Search: id:A117278
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| A117278 |
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Triangle read by rows: T(n,k) is the number of partitions of n into k prime parts (n>=2, 1<=k<=floor(n/2)). |
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+0
1
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| 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 2, 1, 3, 2, 1, 1, 0, 1, 3, 2, 3, 2, 1, 0, 2, 2, 3, 3, 2, 1, 1, 1, 0, 4, 3, 3, 3, 2, 1, 0, 2, 2, 4, 3, 4, 2, 1, 1, 1, 1, 3, 4, 5, 3, 3, 2, 1, 0, 2, 2, 6, 4, 4, 4, 2, 1, 1, 0, 1, 5, 3, 6
(list; graph; listen)
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OFFSET
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2,19
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COMMENT
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Row n has floor(n/2) terms. Row sums yield A000607. T(n,1)=A010051(n) (the characteristic function of the primes). T(n,2)=A061358(n). Sum(k*T(n,k),k>=1) = A084993(n).
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FORMULA
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G.f.=G(t,x)=-1+1/product(1-tx^(p(j)), j=1..infinity), where p(j) is the j-th prime.
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EXAMPLE
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T(12,3)=2 because we have [7,3,2] and [5,5,2].
Triangle starts:
1;
1;
0,1;
1,1;
0,1,1;
1,1,1;
0,1,1,1;
0,1,2,1;
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MAPLE
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g:=1/product(1-t*x^(ithprime(j)), j=1..30): gser:=simplify(series(g, x=0, 30)): for n from 2 to 22 do P[n]:=sort(coeff(gser, x^n)) od: for n from 2 to 22 do seq(coeff(P[n], t^j), j=1..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000607, A010051, A061358, A084993.
Sequence in context: A024361 A135486 A030187 this_sequence A140082 A025852 A025846
Adjacent sequences: A117275 A117276 A117277 this_sequence A117279 A117280 A117281
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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), Mar 07 2006
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