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Search: id:A116685
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| A116685 |
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Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0). |
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+0
3
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| 1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 2, 3, 1, 1, 2, 5, 3, 3, 1, 1, 4, 4, 6, 3, 3, 1, 1, 3, 6, 6, 7, 3, 3, 1, 1, 4, 6, 10, 7, 7, 3, 3, 1, 1, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 4, 10, 21, 22, 24, 17, 14, 8, 7, 3, 3, 1, 1, 4, 11, 21
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Also number of partitions of n such that the difference between the largest and smallest parts is k (see A097364). Example: T(6,2)=3 because we have [4,2],[3,2,1], and [3,1,1,1]. Row 1 has one term; row n (n>=2) has n-1 terms. Row sums yield the partition numbers (A000041). T(n,0)=A000005(n) (number of divisors of n). T(n,1)=A049820(n) (n minus number of divisors of n). T(n,2)=A008805(n-4) for n>=4. Sum(k*T(n,k),k=0..n-2)=A116686 The same as A097364 without the 0's.
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FORMULA
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G.f.=sum(x^i/[(1-x^i)*product(1-tx^j,j=1..i-1), i=1..infinity)]
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EXAMPLE
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T(6,2)=3 because we have [4,1,1],[3,2,1], and [2,2,1,1].
Triangle starts:
1;
2;
2,1;
3,1,1;
2,3,1,1;
4,2,3,1,1;
2,5,3,3,1,1;
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MAPLE
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g:=sum(x^i/(1-x^i)/product(1-t*x^j, j=1..i-1), i=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..n-2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A000005, A049820, A008805, A116686, A097364.
Adjacent sequences: A116682 A116683 A116684 this_sequence A116686 A116687 A116688
Sequence in context: A120967 A116687 A056044 this_sequence A051135 A135352 A072528
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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 23 2006
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