A235998 Triangle read by rows: T(n,k) is the number of compositions of n having k distinct parts (n>=1, 1<=k<=floor((sqrt(1+8*n)-1)/2)).

1, 2, 2, 2, 3, 5, 2, 14, 4, 22, 6, 2, 44, 18, 4, 68, 56, 3, 107, 146, 4, 172, 312, 24, 2, 261, 677, 84, 6, 396, 1358, 288, 2, 606, 2666, 822, 4, 950, 5012, 2226, 4, 1414, 9542, 5304, 120, 5, 2238, 17531, 12514, 480, 2, 3418, 32412, 27904, 1800, 6, 5411, 58995, 61080, 5580

Offset: 1

Author

Omar E. Pol, Jan 19 2014

Keywords

• nonn
• tabf
• nice
• look

Comments

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

The equivalent sequence for partitions is A116608.

For the number of compositions of n see A011782.

For the connection to overcompositions see A235999.

Row sums give A011782(n), n >= 1.

First column is A000005, second column is A131661.

T(k*(k+1)/2,k) = T(A000217(k),k) = A000142(k) = k!. - Alois P. Heinz, Jan 20 2014

Examples

			Triangle begins:
  1;
  2;
  2,    2;
  3,    5;
  2,   14;
  4,   22,     6;
  2,   44,    18;
  4,   68,    56;
  3,  107,   146;
  4,  172,   312,    24;
  2,  261,   677,    84;
  6,  396,  1358,   288;
  2,  606,  2666,   822;
  4,  950,  5012,  2226;
  4, 1414,  9542,  5304,  120;
  5, 2238, 17531, 12514,  480;
  2, 3418, 32412, 27904, 1800;
  6, 5411, 58995, 61080, 5580;
  ...
		

Links

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

Crossrefs

Cf. A003056, A116608, A235790, A235999, A236002.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..25); # Alois P. Heinz, Jan 20 2014, revised May 25 2014

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Sum[b[n-i*j, i-1, p+ j]/j!*If[j==0, 1, x], {j, 0, n/i}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* Jean-François Alcover, Dec 10 2015, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jan 19 2014