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

1,2

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.

Links

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

Formula

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

Example

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;
  ...

Maple

A003056 := proc(n)
    floor((sqrt(1+8*n)-1)/2) ;
end proc:
nDiffParts := proc(L)
    nops(convert(L, set)) ;
end proc:
A116608 := proc(n, k)
    local a, L;
    a :=0 ;
    for L in combinat[partition](n) do
        if nDiffParts(L) = k then
            a := a+1 ;
        end  if;
    end do:
    a ;
end proc:
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 *)

Crossrefs

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

Sequence in context: A198113 A264962 A082193 * A274926 A079171 A029678

Adjacent sequences: A255764 A255765 A255766 * A255768 A255769 A255770

Keyword

nonn,tabf,look

Author

Omar E. Pol, May 21 2015

Extensions

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

Status

approved