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Search: id:A116927
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| A116927 |
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Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having k 1's (n>=1,k>=0). |
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+0
2
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| 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 2, 0, 2, 0, 1, 0, 1, 0, 1, 2, 1, 1
(list; graph; listen)
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OFFSET
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1,60
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COMMENT
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Row 1 has 2 terms; row 2 has one term; row 2n-1 has n terms; row 2n has n-1 terms. Row sums yield A000700. Column 0, except for the first term, yields A090723. Sum(k*T(n,k),k>=0)=A116928(n).
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FORMULA
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G.f.=tx-x+sum(x^(k^2)/(1-tx^2)/product(1-x^(2j), j=2..k), k=1..infinity).
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EXAMPLE
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T(22,3)=2 because we have [8,5,2,2,2,1,1,1] and [7,4,4,4,1,1,1].
Triangle starts:
0,1;
0;
0,1;
1;
0,0,1;
0,1;
0,0,0,1;
1,0,1;
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MAPLE
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g:=t*x-x+sum(x^(k^2)/(1-t*x^2)/product(1-x^(2*j), j=2..k), k=1..30): gser:=simplify(series(g, x=0, 35)): for n from 1 to 30 do P[n]:=sort(coeff(gser, x^n)) od: d:=proc(n) if n=1 then 1 elif n=2 then 0 elif n mod 2 = 1 then (n-1)/2 else (n-4)/2 fi end: for n from 1 to 30 do seq(coeff(P[n], t, j), j=0..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]
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CROSSREFS
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Cf. A000700, A090723, A116928.
Sequence in context: A106404 A083889 A127523 this_sequence A137276 A140581 A137277
Adjacent sequences: A116924 A116925 A116926 this_sequence A116928 A116929 A116930
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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 26 2006
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