A213947 Triangle read by rows: columns are finite differences of the INVERT transform of (1, 2, 3, ...) terms.

1, 1, 2, 1, 4, 3, 1, 10, 6, 4, 1, 20, 21, 8, 5, 1, 42, 57, 28, 10, 6, 1, 84, 150, 88, 35, 12, 7, 1, 170, 390, 252, 110, 42, 14, 8, 1, 340, 990, 712, 335, 132, 49, 16, 9, 1, 682, 2475, 1992, 975, 402, 154, 56, 18, 10

Offset: 1

Author

Gary W. Adamson, Jun 25 2012

Keywords

• nonn
• tabl

Comments

Create an array in which the n-th row is the output of the INVERT transform on the first n natural numbers followed by zeros:

1, 1, 1, 1, 1, 1, 1, ...

1, 3, 5, 11, 21, 43, 85, ... (A001045)

1, 3, 8, 17, 42, 100, 235, ... (A101822)

1, 3, 8, 21, 50, 128, 323, ...

...

For example, row 3 is the INVERT transform of (1, 2, 3, 0, 0, 0, ...). Then, take finite differences of column terms starting from the top; which become the rows of the triangle.

Examples

			First few rows of the triangle:
  1;
  1,    2;
  1,    4,    3;
  1,   10,    6,    4;
  1,   20,   21,    8,    5;
  1,   42,   57,   28,   10,    6;
  1,   84,  150,   88,   35,   12,   7;
  1,  170,  390,  252,  110,   42,  14,   8;
  1,  340,  990,  712,  335,  132,  49,  16,  9;
  1,  682, 2475, 1992,  975,  402, 154,  56, 18, 10;
  1, 1364, 6138, 5464, 2805, 1200, 469, 176, 63, 20, 11;
  ...
		

Crossrefs

Cf. A001906 (row sums), A026644 (2nd column).

Programs

Maple

read("transforms") ;
A213947i := proc(n,k)
        L := [seq(i,i=1..n),seq(0,i=0..k)] ;
        INVERT(L) ;
        op(k,%) ;
end proc:
A213947 := proc(n,k)
        if k = 1 then
                1;
        else
        A213947i(k,n)-A213947i(k-1,n) ;
        end if;
end proc: # R. J. Mathar, Jun 30 2012