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Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.
13

%I #39 Oct 20 2016 10:25:02

%S 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 200,32,12,116,296,80,4,148,416,160,8,176,536,320,8,176,776,480,32,10,

%U 220,936,832,64,4,236,1232,1232,160,12,272,1472,1872,320

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

%C 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 Row n has length A003056(n) hence the first element of column k is in row A000217(k).

%C The first element of column k is A000079(k).

%H Alois P. Heinz, <a href="/A235790/b235790.txt">Rows n = 1..500, flattened</a>

%e Triangle begins:

%e 2;

%e 4;

%e 4, 4;

%e 6, 8;

%e 4, 20;

%e 8, 24, 8;

%e 4, 44, 16;

%e 8, 52, 40;

%e 6, 68, 80;

%e 8, 88, 120, 16;

%e 4, 108, 200, 32;

%e 12, 116, 296, 80;

%e 4, 148, 416, 160;

%e 8, 176, 536, 320;

%e 8, 176, 776, 480, 32;

%e 10, 220, 936, 832, 64;

%e 4, 236, 1232, 1232, 160;

%e 12, 272, 1472, 1872, 320;

%e 4, 284, 1880, 2592, 640;

%e 12, 324, 2216, 3632, 1152;

%e 8, 328, 2704, 4944, 1856, 64;

%e ...

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p expand(b(n, i-1)+add(x*b(n-i*j, i-1), j=1..n/i))))

%p end:

%p T:= n->(p->seq(2^i*coeff(p, x, i), i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=1..20); # _Alois P. Heinz_, Jan 20 2014

%t 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 Row sums give A015128, n >= 1.

%Y Column 1 is A062011.

%Y Cf. A000217, A003056, A116608, A196020, A211971, A235792, A235793, A235797, A235798, A235999, A236000, A236001.

%K nonn,tabf,look

%O 1,1

%A _Omar E. Pol_, Jan 18 2014