A116856 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts such that the smallest part is k (n>=1, k>=1).

1, 1, 1, 0, 1, 2, 2, 0, 0, 0, 1, 3, 0, 1, 4, 0, 0, 0, 0, 0, 1, 5, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 8, 0, 1, 0, 1, 10, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 12, 0, 2, 0, 1, 15, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18, 0, 2, 0, 1, 0, 1, 22, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 0, 3, 0, 1, 0, 1, 32, 0, 4, 0, 1, 0

Offset: 1

Author

Emeric Deutsch, Feb 24 2006

Keywords

• nonn
• tabf

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.

Examples

			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
		

Crossrefs

Cf. A000009, A092314, A116799.

Programs

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 *)

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).