A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 18, 97, 180, 100, 1, 35, 403, 1829, 3160, 1700, 1, 61, 1313, 12307, 50714, 83860, 44200, 1, 98, 3570, 60888, 506073, 1960278, 3147020, 1635400, 1, 148, 8470, 239388, 3550473, 27263928, 101160920, 158986400, 81770000, 1, 213

Offset: 1

Author

Clark Kimberling, Sep 11 2024

Keywords

• nonn
• tabf

Comments

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

Examples

			First 3 polynomials in s(x)**t(x) are
  1 + x,
  1 + 3 x + 2 x^2,
  1 + 8 x + 17 x^2 + 10 x^3.
First 5 rows of array:
  1    1
  1    3     2
  1    8    17    10
  1   18    97   180   100
  1   35  4034  1829  3160  1700
		

Crossrefs

Cf. A000290, A081489 (column 2), A101686 (T(n,n+1)), A374848, A375042, A375043.

Programs

Mathematica

s[n_] := n^2  x; t[n_] := 1 + x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)