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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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1
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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; refs; listen; history; internal format)
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OFFSET
| 1,60
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COMMENTS
| 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
| 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
| 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
| 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
| Cf. A000700, A090723, A116928.
Sequence in context: A106404 A083889 A127523 * A137276 A140581 A137277
Adjacent sequences: A116924 A116925 A116926 * A116928 A116929 A116930
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 26 2006
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