A136406 - OEIS

%I #9 Oct 23 2019 16:28:05

%S 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,

%T 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,

%U 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

%N 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.

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

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

%H Andrew Howroyd, <a href="/A136406/b136406.txt">Table of n, a(n) for n = 1..1275</a>

%e Triangle begins:

%e   1,

%e   1, 1;

%e   1, 1,  1;

%e   1, 3,  1,  1;

%e   1, 2,  3,  1,  1;

%e   1, 3,  4,  3,  1,  1;

%e   1, 3,  5,  4,  3,  1, 1;

%e   1, 5,  6,  8,  4,  3, 1, 1;

%e   1, 4, 10,  8,  8,  4, 3, 1, 1;

%e   1, 5, 10, 14, 11,  8, 4, 3, 1, 1;

%e   1, 5, 12, 16, 17, 11, 8, 4, 3, 1, 1;

%e   ...

%o (PARI)

%o P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}

%o 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)}

%o { for(n=1, 10, print(T(n))) } \\ _Andrew Howroyd_, Oct 23 2019

%Y Row sums are A004101.

%Y Cf. A136405, A137504.

%K nonn,tabl

%O 1,8

%A _Benoit Jubin_, Apr 13 2008

%E Terms a(68) and beyond from _Andrew Howroyd_, Oct 22 2019