A375042 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) = 2x+1. See Comments.

1, 2, 1, 5, 6, 1, 11, 36, 36, 1, 22, 157, 432, 396, 1, 40, 553, 3258, 8172, 7128, 1, 67, 1633, 18189, 96138, 227772, 192456, 1, 105, 4179, 80243, 787320, 3881016, 8847792, 7313328, 1, 156, 9534, 293372, 4879713, 44034336, 206779608, 458550720, 372979728, 1

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 + 2x,
  1 + 5 x + 6 x^2,
  1 + 11 x + 36 x^2 + 36 x^3.
First 5 rows of array:
  1   2
  1   5      6
  1   11    36    36
  1   22   157   432   396
  1   40   553  3258  8172  7128
		

Crossrefs

Cf. A000290, A277355 ((1/2)T(n,n+1)), A374848, A375041, A375043.

Programs

Mathematica

s[n_] := n^2  x; t[n_] := 1 + 2 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 *)