A255767 Triangle read by rows: T(n,k) = sum of all parts of all partitions of n into k distinct parts.

1, 4, 6, 3, 12, 8, 10, 25, 24, 36, 6, 14, 77, 14, 32, 104, 40, 27, 153, 90, 40, 220, 150, 10, 22, 297, 275, 22, 72, 348, 444, 60, 26, 481, 676, 130, 56, 616, 938, 280, 60, 660, 1455, 450, 15, 80, 880, 1872, 832, 32, 34, 1003, 2618, 1309, 85, 108, 1224, 3312, 2106, 180, 38, 1349, 4465, 3078, 380, 120, 1620, 5540, 4540, 720

Offset: 1

Author

Omar E. Pol, May 21 2015

Keywords

• nonn
• tabf
• look

Comments

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

The first element of column k is A000217(k).

Column 1 is A038040.

Examples

			Triangle begins:
    1;
    4;
    6,    3;
   12,    8;
   10,   25;
   24,   36,    6;
   14,   77,   14;
   32,  104,   40;
   27,  153,   90;
   40,  220,  150,   10;
   22,  297,  275,   22;
   72,  348,  444,   60;
   26,  481,  676,  130;
   56,  616,  938,  280;
   60,  660, 1455,  450,  15;
   80,  880, 1872,  832,  32;
   34, 1003, 2618, 1309,  85;
  108, 1224, 3312, 2106, 180;
   38, 1349, 4465, 3078, 380;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..500, flattened

Crossrefs

Cf. A000217, A003056, A038040, A066186 (row sums), A116608, A255768.

Programs

Maple

A255767 := proc(n,k)
    n*A116608(n,k) ;
end proc:
for n from 1 to 20 do
    for k from 1 to A003056(n) do
        printf("%d,",A255767(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 10 2015
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)
      ->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, n*[b(n, n)])[]:
seq(T(n), n=1..30);  # Alois P. Heinz, Jul 26 2015

Mathematica

nmax = 30; T = Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[-1 + Product[1 + t x^j/(1 - x^j), {j, 1, nmax}] + O[x]^(nmax+1), x]];
Table[n*T[[n]], {n, 1, nmax}] // Flatten (* Jean-François Alcover, May 19 2018 *)

Formula

T(n,k) = n * A116608(n,k).

Extensions

a(7) and beyond from R. J. Mathar, Jul 10 2015