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A255767 Triangle read by rows: T(n,k) = sum of all parts of all partitions of n into k distinct parts. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified February 19 00:35 EST 2020. Contains 332028 sequences. (Running on oeis4.)