A168532 Triangle read by rows, A054525 * A168021.

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 14, 0, 0, 0, 0, 0, 1, 17, 3, 0, 1, 0, 0, 0, 1, 27, 0, 2, 0, 0, 0, 0, 0, 1, 34, 6, 0, 0, 1, 0, 0, 0, 0, 1, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 63, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Nov 28 2009

Keywords

• nonn
• tabl

Comments

Row sums = A000041 starting (1, 2, 3, 5, 7, 11, 15, ...).

T(n,k) is the number of partitions of n into parts with GCD = k. - Alois P. Heinz, Jun 06 2013

Examples

			First few rows of the triangle:
    1;
    1,  1;
    2,  0, 1;
    3,  1, 0, 1;
    6,  0, 0, 0, 1;
    7,  2, 1, 0, 0, 1;
   14,  0, 0, 0, 0, 0, 1;
   17,  3, 0, 1, 0, 0, 0, 1;
   27,  0, 2, 0, 0, 0, 0, 0, 1;
   34,  6, 0, 0, 1, 0, 0, 0, 0, 1;
   55,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   63,  7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
  100,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  119, 14, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
  167,  0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  209, 17, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  296,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Cf. A168021, A000837.

Cf. A256067 (the same for LCM).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^igcd(t, i),
      t=0..degree(p)))(add(b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..17);  # Alois P. Heinz, Mar 29 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i==1, x, b[n, i-1] + Function[{p}, Sum[Coefficient[p, x, t]*x^GCD[t, i], {t, 0, Exponent[p, x]}]][Sum[b[n - i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, n]]; Table[T[n], {n, 1, 17}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

Formula

Mobius transform of triangle A168021 = an infinite lower triangular matrix with aerated variants of A000837 in each column; where A000837 = the Mobius transform of the partition numbers, A000041.

Extensions

Corrected and extended by Alois P. Heinz, Jun 06 2013