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Search: id:A116608
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| A116608 |
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Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1). |
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+0
1
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| 1, 2, 2, 1, 3, 2, 2, 5, 4, 6, 1, 2, 11, 2, 4, 13, 5, 3, 17, 10, 4, 22, 15, 1, 2, 27, 25, 2, 6, 29, 37, 5, 2, 37, 52, 10, 4, 44, 67, 20, 4, 44, 97, 30, 1, 5, 55, 117, 52, 2, 2, 59, 154, 77, 5, 6, 68, 184, 117, 10, 2, 71, 235, 162, 20, 6, 81, 277, 227, 36, 4, 82, 338, 309, 58, 1, 4, 102
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number). Row sums yield the partition numbers (A000041). T(n,1)=A000005(n) (number of divisors of n). T(n,2)=A002133(n). T(n,3)=A002134(n). Sum(k*T(n,k), k>=1)=A000070(n-1).
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FORMULA
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G.f.=-1+product(1 + tx^j/(1-x^j), j=1..infinity).
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EXAMPLE
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T(6,2)=6 because we have [5,1],[4,2],[4,1,1],[3,1,1,1],[2,2,1,1], and [2,1,1,1,1,1] ([6],[3,3],[3,2,1],[2,2,2], and [1,1,1,1,1,1] do not qualify).
Triangle starts:
1;
2;
2,1;
3,2;
2,5;
4,6,1;
2,11,2;
4,13,5;
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MAPLE
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g:=product(1+t*x^j/(1-x^j), j=1..30)-1: gser:=simplify(series(g, x=0, 27)): for n from 1 to 23 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 23 do seq(coeff(P[n], t^j), j=1..floor(sqrt(1+8*n)/2-1/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A000005, A002133, A002134, A000070.
Adjacent sequences: A116605 A116606 A116607 this_sequence A116609 A116610 A116611
Sequence in context: A112221 A130816 A109951 this_sequence A002947 A128180 A074754
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 19 2006
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