A375049 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) = x+F(n) and t(x) = x+F(n), and F(n) = n-th Fibonacci number (A000045). See Comments.

0, 2, 1, 4, 4, 2, 10, 16, 8, 16, 64, 96, 64, 16, 162, 594, 864, 624, 224, 32, 3600, 11280, 14596, 9984, 3808, 768, 64, 147456, 393216, 443392, 273920, 100096, 21632, 2560, 128, 12320100, 27335880, 26086356, 13971408, 4589488, 946176, 119488, 8448, 256

Offset: 1

Author

Clark Kimberling, Jul 31 2024

Keywords

• nonn
• tabf

Comments

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays. If n is odd, then the polynomial u(n) is a square. Every T(n,k) except T(2,1) is even.

Examples

			First 3 polynomials in s(x)**t(x) are
  0 + 2x,
  1 + 4 x + 4x^2,
  2 + 10 x + 16 x^2 + 8 x^3.
First 5 rows of array:
  0   2
  1   4   4
  2   10  16   8
  16  64  96  64  16
  162 594 864 624 224 32
		

Crossrefs

Cf. A000045, A000079 (T(n,n+1)), A374848, A375047, A375048.

Programs

Mathematica

s[n_] := x + Fibonacci[n]; t[n_] := Fibonacci[n];
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 *)