A136406 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i^2 = n.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 4, 3, 1, 1, 1, 5, 6, 8, 4, 3, 1, 1, 1, 4, 10, 8, 8, 4, 3, 1, 1, 1, 5, 10, 14, 11, 8, 4, 3, 1, 1, 1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1, 1, 7, 14, 23, 21, 21, 11, 8, 4, 3, 1, 1, 1, 6, 17, 25, 32, 24, 21, 11, 8, 4, 3, 1, 1

Offset: 1

Author

Benoit Jubin, Apr 13 2008

Keywords

• nonn
• tabl

Comments

T(n,1) = T(n,n) = 1.

T(n,n-k) does not depend on k if k <= floor(n/2).

Examples

			Triangle begins:
  1,
  1, 1;
  1, 1,  1;
  1, 3,  1,  1;
  1, 2,  3,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  4,  3,  1, 1;
  1, 5,  6,  8,  4,  3, 1, 1;
  1, 4, 10,  8,  8,  4, 3, 1, 1;
  1, 5, 10, 14, 11,  8, 4, 3, 1, 1;
  1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums are A004101.

Cf. A136405, A137504.

Programs

PARI

P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, sqrtint(n), sum(k=0, n\w^2, (x^w*y)^(k*w) / P(k,w^2,n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019

Extensions

Terms a(68) and beyond from Andrew Howroyd, Oct 22 2019