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).
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
Offset: 1
Examples
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;
Programs
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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]
Formula
G.f.=tx-x+sum(x^(k^2)/(1-tx^2)/product(1-x^(2j), j=2..k), k=1..infinity).
Comments