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A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.

%I A235790 #39 Oct 20 2016 10:25:02
%S A235790 2,4,4,4,6,8,4,20,8,24,8,4,44,16,8,52,40,6,68,80,8,88,120,16,4,108,
%T A235790 200,32,12,116,296,80,4,148,416,160,8,176,536,320,8,176,776,480,32,10,
%U A235790 220,936,832,64,4,236,1232,1232,160,12,272,1472,1872,320
%N A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.
%C A235790 It appears that T(n,k) is the number of overpartitions of n having k distinct parts. (This is true by definition, _Joerg Arndt_, Jan 20 2014).
%C A235790 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C A235790 The first element of column k is A000079(k).
%H A235790 Alois P. Heinz, <a href="/A235790/b235790.txt">Rows n = 1..500, flattened</a>
%e A235790 Triangle begins:
%e A235790 2;
%e A235790 4;
%e A235790 4,    4;
%e A235790 6,    8;
%e A235790 4,   20;
%e A235790 8,   24,    8;
%e A235790 4,   44,   16;
%e A235790 8,   52,   40;
%e A235790 6,   68,   80;
%e A235790 8,   88,  120,   16;
%e A235790 4,  108,  200,   32;
%e A235790 12, 116,  296,   80;
%e A235790 4,  148,  416,  160;
%e A235790 8,  176,  536,  320;
%e A235790 8,  176,  776,  480,   32;
%e A235790 10, 220,  936,  832,   64;
%e A235790 4,  236, 1232, 1232,  160;
%e A235790 12, 272, 1472, 1872,  320;
%e A235790 4,  284, 1880, 2592,  640;
%e A235790 12, 324, 2216, 3632, 1152;
%e A235790 8,  328, 2704, 4944, 1856, 64;
%e A235790 ...
%p A235790 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A235790       expand(b(n, i-1)+add(x*b(n-i*j, i-1), j=1..n/i))))
%p A235790     end:
%p A235790 T:= n->(p->seq(2^i*coeff(p, x, i), i=1..degree(p)))(b(n$2)):
%p A235790 seq(T(n), n=1..20);  # _Alois P. Heinz_, Jan 20 2014
%t A235790 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[b[n, i-1] + Sum[x*b[n-i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[p, Table[2^i * Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Oct 20 2016, after _Alois P. Heinz_ *)
%Y A235790 Row sums give A015128, n >= 1.
%Y A235790 Column 1 is A062011.
%Y A235790 Cf. A000217, A003056, A116608, A196020, A211971, A235792, A235793, A235797, A235798, A235999, A236000, A236001.
%K A235790 nonn,tabf,look
%O A235790 1,1
%A A235790 _Omar E. Pol_, Jan 18 2014