1,2

Leftmost column of the triangle = pentagonal numbers (A000326). Rightmost diagonal = factorial numbers).

Given polynomials with coefficients (1, 4, 7...); i.e. (1); (n + 4); (n^2 + 4n + 7);...; we generate an array by rows, f(n). Inverse binomial transforms of row n of the array become row n of the triangle.

First few rows of the array are:

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

5, 6, 7, 8, 9,...

12, 19, 28, 39, 52,...

...

such that for example, x^n + 4n + 7 generates (12, 19, 28, 39, 52...); with inverse binomial transform of that sequence becoming row 3 of the triangle: (12, 7, 2).

First few rows of the triangle are:

1,

5, 1;

12, 7, 2;

22, 26, 20, 6;

35, 74, 112, 84, 24;

51, 183, 484, 672, 456, 120;

...

Cf. A000326.

Sequence in context: A063004 A104572 A125232 this_sequence A062264 A094049 A081224

Adjacent sequences: A116920 A116921 A116922 this_sequence A116924 A116925 A116926

nonn,tabl,uned

Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 26 2006