A116679 Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k even parts (n >= 0, k >= 0).

1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 3, 1, 2, 4, 2, 2, 5, 3, 2, 6, 4, 3, 7, 4, 1, 3, 8, 6, 1, 3, 10, 8, 1, 4, 11, 10, 2, 5, 13, 11, 3, 5, 15, 14, 4, 5, 18, 18, 5, 6, 20, 21, 7, 7, 23, 24, 9, 1, 8, 26, 29, 12, 1, 8, 30, 36, 14, 1, 9, 34, 41, 18, 2, 11, 38, 47, 23, 3, 12, 43, 55, 28, 4

Offset: 0

Author

Emeric Deutsch, Feb 22 2006

Keywords

• nonn
• tabf

Comments

Row n contains floor((1 + sqrt(1+4*n))/2) terms.

Row sums yield A000009.

T(n,0) = A000700(n), T(n,1) = A096911(n) for n >= 1.

Sum_{k>=0} k*T(n,k) = A116680(n).

Examples

			T(9,2)=2 because we have [6,2,1] and [4,3,2].
Triangle starts:
  1;
  1;
  0, 1;
  1, 1;
  1, 1;
  1, 2;
  1, 2, 1;
  1, 3, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000009, A000700, A096911, A116680.

Programs

Maple

g:=product((1+x^(2*j-1))*(1+t*x^(2*j)),j=1..25): gser:=simplify(series(g,x=0,38)): P[0]:=1: for n from 1 to 27 do P[n]:=sort(coeff(gser,x^n)) od: for n from 0 to 27 do seq(coeff(P[n],t,j),j=0..floor((sqrt(1+4*n)-1)/2)) od; # yields sequence in triangular form

Mathematica

With[{m=25}, CoefficientList[CoefficientList[Series[Product[(1+x^(2*j- 1))*(1+t*x^(2*j)), {j,1,m+2}], {x,0,m}, {t,0,m}], x], t]]//Flatten (* G. C. Greubel, Jun 07 2019 *)

Formula

G.f.: Product_{j>=1} (1+x^(2*j-1))*(1+t*x^(2*j)).