A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1

Offset: 0

Author

Omar E. Pol, Jun 27 2012

Keywords

• nonn
• tabl

Comments

It appears that row 2 is A034856.

Observation:

Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...

Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

Examples

			Array begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
1,   4,   8,  13,  19,  26,  34,  43,  53,
1,   7,  18,  35,  59,  91, 132, 183,
1,  12,  38,  86, 164, 281, 447,
1,  19,  74, 194, 416, 787,
1,  30, 139, 415, 990,
1,  45, 249, 844,
1,  67, 434,
1,  97,
1,
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns (0-3): A000012, A000070, A000713, A210843.

Rows (0-1): A000012, A000027.

Main diagonal gives A303070.

Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

Programs

Maple

with(numtheory):
etr:= proc(p) local b;
        b:= proc(n) option remember; `if`(n=0, 1,
              add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
            end
      end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013

Mathematica

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)