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A116927 Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having k 1's (n>=1,k>=0).

%I A116927 #3 Mar 30 2012 17:36:08
%S A116927 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,
%T A116927 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,
%U A116927 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
%N A116927 Triangle read by rows: T(n,k) is the number of self-conjugate partitions of n having k 1's (n>=1,k>=0).
%C A116927 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).
%F A116927 G.f.=tx-x+sum(x^(k^2)/(1-tx^2)/product(1-x^(2j), j=2..k), k=1..infinity).
%e A116927 T(22,3)=2 because we have [8,5,2,2,2,1,1,1] and [7,4,4,4,1,1,1].
%e A116927 Triangle starts:
%e A116927 0,1;
%e A116927 0;
%e A116927 0,1;
%e A116927 1;
%e A116927 0,0,1;
%e A116927 0,1;
%e A116927 0,0,0,1;
%e A116927 1,0,1;
%p A116927 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]
%Y A116927 Cf. A000700, A090723, A116928.
%K A116927 nonn,tabf
%O A116927 1,60
%A A116927 _Emeric Deutsch_, Feb 26 2006