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Search: id:A118198
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| A118198 |
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Triangle read by rows: T(n,k) is the number of partitions of n having k parts equal to the size of the Durfee square (0<=k<=n). |
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2
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| 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 2, 1, 0, 1, 2, 3, 2, 2, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 5, 6, 4, 2, 2, 1, 1, 0, 1, 7, 8, 5, 4, 2, 1, 1, 1, 0, 1, 10, 11, 8, 5, 2, 2, 1, 1, 1, 0, 1, 13, 15, 11, 7, 3, 2, 1, 1, 1, 1, 0, 1, 18, 20, 16, 9, 5, 2, 2, 1, 1, 1, 1, 0, 1, 23, 27, 21, 13, 6, 3, 2, 1
(list; graph; listen)
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OFFSET
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0,13
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COMMENT
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Row sums yield the partition numbers (A000041). T(n,0)=A118199(n)
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FORMULA
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G.f.=G(t,x)=1+sum(x^(k^2)*P(k-1)sum(t^(k-j)*x^j*P(j), j=0..k)/(1-tx^k), k=1..infinity), where P(m)=1/product(1-x^i,i=1..m).
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EXAMPLE
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T(4,0)=1 because [4] has Durfee square of size 1 and there is no part equal to 1; T(4,1)=1 because [3,1] has Durfee square of size 1 and there is 1 part equal to 1; T(4,2)=2 because [2,2] has Durfee square of size 2 and there are 2 parts equal to 2 and [2,1,1] has Durfee square of size 1 and there are 2 parts equal to 1; T(4,3)=0 because obviously no partition of 4 can have exactly 3 parts of the same size; T(4,4)=1 because [1,1,1,1] has Durfee square of size 1 and there are 4 parts equal to 1.
Triangle starts:
1;
0,1;
1,0,1;
1,1,0,1;
1,1,2,0,1;
1,2,2,1,0,1;
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MAPLE
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g:=1+sum(x^(k^2)*sum(t^(k-j)*x^j/product(1-x^i, i=1..j), j=0..k)/(1-t*x^k)/product(1-x^i, i=1..k-1), k=1..20): gser:=simplify(series(g, x=0, 30)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 13 do seq(coeff(P[n], t, p), p=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A118199.
Sequence in context: A077763 A030218 A127440 this_sequence A059881 A096568 A025862
Adjacent sequences: A118195 A118196 A118197 this_sequence A118199 A118200 A118201
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 14 2006
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