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

1,3

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.

Links

Table of n, a(n) for n=1..55.

Example

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;
  ...

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

Crossrefs

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

Sequence in context: A319075 A264871 A067410 * A348482 A188403 A248929

Adjacent sequences: A213944 A213945 A213946 * A213948 A213949 A213950

Keyword

nonn,tabl

Author

Gary W. Adamson, Jun 25 2012

Status

approved