A265249 Triangle read by rows: T(n,k) is the number of partitions of n having k parts strictly between the smallest and the largest part (n>=1, k>=0).

1, 2, 3, 5, 7, 10, 1, 13, 2, 17, 4, 1, 20, 8, 2, 26, 11, 4, 1, 29, 17, 8, 2, 35, 24, 13, 4, 1, 39, 33, 19, 8, 2, 48, 39, 30, 13, 4, 1, 48, 56, 41, 21, 8, 2, 60, 64, 57, 32, 13, 4, 1, 61, 83, 75, 47, 21, 8, 2, 74, 94, 100, 65, 34, 13, 4, 1

Offset: 1

Author

Emeric Deutsch, Dec 25 2015

Keywords

• nonn
• tabf

Comments

Number of entries in row n is floor((n-4)/2) (n>=4).

Sum of entries of row n = A000041(n) = number of partitions of n.

T(n,0) = A265250(n).

Sum(k*T(n,k), k>=0) = A182977(n).

Examples

			T(8,2) = 1 because among the 22 partitions of 8 only [3,2,2,1] has 2 parts strictly between the smallest and the largest part.
Triangle starts:
1;
2;
3;
5;
7;
10, 1;
13, 2;
		

Crossrefs

CF. A000041, A182977, A265250.

Programs

Maple

g := add(x^i/(1-x^i), i=1..80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)*mul(1-t*x^k, k=i+1..j-1)),j=i+1..80),i=1..80): gser := simplify(series(g,x=0,23)): for n to 22 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 22 do seq(coeff(P[n],t,k), k=0..degree(P[n])) end do; # yields sequence in triangular form

Formula

G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/(1-x^i)/(1-x^j)/Product_{k=i+1..j-1} (1-tx^k).