cp's OEIS Frontend

A370005 Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A370005 #26 Feb 11 2024 18:56:04
%S A370005 1,0,1,2,1,1,4,3,1,1,12,7,4,1,1,16,19,8,4,1,1,48,35,23,9,4,1,1,60,83,
%T A370005 43,24,9,4,1,1,148,143,106,47,25,9,4,1,1,220,291,186,115,48,25,9,4,1,
%U A370005 1,438,511,397,210,119,49,25,9,4,1,1,618,949,697,444,219,120,49,25,9,4,1,1
%N A370005 Number T(n,k) of ordered pairs of partitions of n with exactly k common parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A370005 Alois P. Heinz, <a href="/A370005/b370005.txt">Rows n = 0..200, flattened</a>
%e A370005 T(4,0) = 12: (1111,22), (1111,4), (211,4), (22,1111), (22,31), (22,4), (31,22), (31,4), (4,1111), (4,211), (4,22), (4,31).
%e A370005 T(4,1) = 7: (1111,31), (211,22), (211,31), (22,211), (31,1111), (31,211), (4,4).
%e A370005 T(4,2) = 4: (1111,211), (211,1111), (22,22), (31,31).
%e A370005 T(4,3) = 1: (211,211).
%e A370005 T(4,4) = 1: (1111,1111).
%e A370005 Triangle T(n,k) begins:
%e A370005     1;
%e A370005     0,   1;
%e A370005     2,   1,   1;
%e A370005     4,   3,   1,   1;
%e A370005    12,   7,   4,   1,   1;
%e A370005    16,  19,   8,   4,   1,  1;
%e A370005    48,  35,  23,   9,   4,  1,  1;
%e A370005    60,  83,  43,  24,   9,  4,  1, 1;
%e A370005   148, 143, 106,  47,  25,  9,  4, 1, 1;
%e A370005   220, 291, 186, 115,  48, 25,  9, 4, 1, 1;
%e A370005   438, 511, 397, 210, 119, 49, 25, 9, 4, 1, 1;
%e A370005   ...
%p A370005 b:= proc(n, m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A370005       add(add(expand(b(sort([n-i*j, m-i*h])[], i-1)*
%p A370005        x^min(j, h)), h=0..m/i), j=0..n/i)))
%p A370005     end:
%p A370005 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$3)):
%p A370005 seq(T(n), n=0..12);
%Y A370005 Column k=0 gives A054440.
%Y A370005 Row sums and T(2n,n) give A001255.
%Y A370005 Cf. A000041, A000290, A260669, A370207.
%K A370005 nonn,tabl
%O A370005 0,4
%A A370005 _Alois P. Heinz_, Feb 07 2024