OFFSET
1,6
COMMENTS
Row 2n-1 has 2n-1 terms; row 4n+2 has 2n+1 terms; row 4n has 2n-1 terms. Row sums yield A000009.
FORMULA
G.f.: Sum_{j=1..oo} t^(2*j-1)*x^(2*j-1)/Product_{i=j..oo} 1-x^(2*i-1).
T(n,2k)=0.
Sum_{k>=1} k*T(n,k) = A092314(n).
EXAMPLE
T(12,3)=2 because we have [9,3] and [3,3,3,3].
Triangle starts:
1;
1;
1,0,1;
2;
2,0,0,0,1;
3,0,1;
4,0,0,0,0,0,1
MAPLE
g:=sum(t^(2*j-1)*x^(2*j-1)/product(1-x^(2*i-1), i=j..20), j=1..30): gser:=simplify(series(g, x=0, 20)): for n from 1 to 17 do P[n]:=sort(coeff(gser, x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n mod 4 = 2 then n/2 else n/2-1 fi end: for n from 1 to 17 do seq(coeff(P[n], t^j), j=1..d(n)) od; # yields sequence in triangular form; d(n) is the degree of the polynomial P[n]
MATHEMATICA
imax = 18;
Rest@CoefficientList[#, t]& /@ Rest@CoefficientList[Sum[t^(2j-1)*x^(2j-1)/ Product[1 - x^(2i-1), {i, j, imax}], {j, 1, imax}] + O[x]^imax, x] // Flatten (* Jean-François Alcover, Aug 26 2024 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 24 2006
STATUS
approved