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A117456
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Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part to the largest part occurs and the number of parts is k (1<=k<=n).
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1
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 2, 1, 1, 1
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OFFSET
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1,18
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COMMENTS
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Row sums yield A034296. sum(k*T(n,k),k=1..n)=A117457(n).
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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G.f.=G(t,x)=sum(t^j*x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..infinity).
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EXAMPLE
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T(10,5)=3 because we have [3,3,2,1,1],[3,2,2,2,1] and [2,2,2,2,2].
Triangle starts:
1;
1,1;
1,1,1;
1,1,1,1;
1,1,1,1,1;
1,1,2,1,1,1;
1,1,1,2,1,1,1;
1,1,1,2,2,1,1,1;
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MAPLE
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g:=sum(t^j*x^j*product(1+x^i, i=1..j-1)/(1-x^j), j=1..60): gser:=simplify(series(g, x=0, 55)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: for n from 1 to 15 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A034296, A117457.
Sequence in context: A030379 A030392 A165633 * A030621 A120336 A039738
Adjacent sequences: A117453 A117454 A117455 * A117457 A117458 A117459
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Mar 18 2006
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STATUS
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approved
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