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%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
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