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A370207 Number T(n,k) of unordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A370207 #21 Feb 11 2024 18:55:25
%S A370207 1,0,1,1,1,1,2,2,1,1,6,4,3,1,1,8,10,5,3,1,1,24,18,13,6,3,1,1,30,42,23,
%T A370207 14,6,3,1,1,74,72,55,26,15,6,3,1,1,110,146,95,61,27,15,6,3,1,1,219,
%U A370207 256,201,109,64,28,15,6,3,1,1,309,475,351,227,115,65,28,15,6,3,1,1
%N A370207 Number T(n,k) of unordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A370207 Alois P. Heinz, <a href="/A370207/b370207.txt">Rows n = 0..200, flattened</a>
%F A370207 T(n,k) = (A370005(n,k) + A072233(n,k))/2.
%e A370207 T(4,0) = 6: (1111,22), (1111,4), (211,4), (22,31), (22,4), (31,4).
%e A370207 T(4,1) = 4: (1111,31), (211,22), (211,31), (4,4).
%e A370207 T(4,2) = 3: (1111,211), (22,22), (31,31).
%e A370207 T(4,3) = 1: (211,211).
%e A370207 T(4,4) = 1: (1111,1111).
%e A370207 Triangle T(n,k) begins:
%e A370207     1;
%e A370207     0,   1;
%e A370207     1,   1,   1;
%e A370207     2,   2,   1,   1;
%e A370207     6,   4,   3,   1,  1;
%e A370207     8,  10,   5,   3,  1,  1;
%e A370207    24,  18,  13,   6,  3,  1,  1;
%e A370207    30,  42,  23,  14,  6,  3,  1, 1;
%e A370207    74,  72,  55,  26, 15,  6,  3, 1, 1;
%e A370207   110, 146,  95,  61, 27, 15,  6, 3, 1, 1;
%e A370207   219, 256, 201, 109, 64, 28, 15, 6, 3, 1, 1;
%e A370207   ...
%p A370207 b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A370207       add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*
%p A370207        x^min(j, h)), h=0..m/i), j=0..n/i)))
%p A370207     end:
%p A370207 g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A370207       add(expand(g(n-i*j, i-1)*x^j), j=0..n/i)))
%p A370207     end:
%p A370207 T:= (n, k)-> (coeff(b(n$3), x, k)+coeff(g(n$2), x, k))/2:
%p A370207 seq(seq(T(n, k), k=0..n), n=0..12);
%Y A370207 Column k=0 gives A260669.
%Y A370207 Row sums and T(2n,n) give A086737.
%Y A370207 Cf. A000041, A000217, A008284, A072233, A370005.
%K A370207 nonn,tabl
%O A370207 0,7
%A A370207 _Alois P. Heinz_, Feb 11 2024