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A136405 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 = n.

%I A136405 #9 Oct 23 2019 16:42:54
%S A136405 1,1,2,1,1,3,1,3,2,5,1,2,4,3,7,1,4,6,7,5,11,1,3,7,8,11,7,15,1,5,8,16,
%T A136405 14,17,11,22,1,4,12,14,23,21,25,15,30,1,6,12,24,29,38,33,37,22,42,1,5,
%U A136405 15,24,41,42,57,47,52,30,56,1,7,18,37,47,75,68,87,70,74,42,77
%N A136405 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 = n.
%H A136405 Andrew Howroyd, <a href="/A136405/b136405.txt">Table of n, a(n) for n = 1..1275</a>
%F A136405 T(n,1) = 1.
%F A136405 T(n,2) = A028242(n).
%F A136405 T(n,n) = A000041(n).
%e A136405 Triangle begins:
%e A136405   1;
%e A136405   1, 2;
%e A136405   1, 1,  3;
%e A136405   1, 3,  2,  5;
%e A136405   1, 2,  4,  3,  7;
%e A136405   1, 4,  6,  7,  5, 11;
%e A136405   1, 3,  7,  8, 11,  7, 15;
%e A136405   1, 5,  8, 16, 14, 17, 11, 22;
%e A136405   1, 4, 12, 14, 23, 21, 25, 15, 30;
%e A136405   1, 6, 12, 24, 29, 38, 33, 37, 22, 42;
%e A136405   ...
%e A136405 T(4,2) = 3 since (4,2) can be bi-partitioned as (2,2) or ((1,1),(3,1)) or ((2,1),(2,1)).
%e A136405 T(5,3) = 4 since (5,3) can be bi-partitioned as ((1,1),(2,2)) or ((3,1),(1,2)) or ((1,1),(1,1),(3,1)) or ((1,1),(2,1),(2,1)).
%o A136405 (PARI)
%o A136405 P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
%o A136405 T(n)={Vecrev(polcoef(prod(w=1, n, sum(k=0, n\w, (x*y)^(k*w)/P(k,w,n))), n)/y)}
%o A136405 { for(n=1, 10, print(T(n))) } \\ _Andrew Howroyd_, Oct 23 2019
%Y A136405 Row sums are A006171.
%Y A136405 Cf. A000041, A028242, A090806, A136406.
%K A136405 nonn,tabl
%O A136405 1,3
%A A136405 _Benoit Jubin_, Apr 13 2008
%E A136405 Terms a(57) and beyond from _Andrew Howroyd_, Oct 23 2019