A375047 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+1 and t(x) = F(n) = n-th Fibonacci number. See Comments.

1, 1, 2, 3, 1, 4, 8, 5, 1, 12, 28, 23, 8, 1, 48, 124, 120, 55, 12, 1, 288, 792, 844, 450, 127, 18, 1, 2592, 7416, 8388, 4894, 1593, 289, 27, 1, 36288, 106416, 124848, 76904, 27196, 5639, 667, 41, 1, 798336, 2377440, 2853072, 1816736, 675216, 151254, 20313

Offset: 1

Author

Clark Kimberling, Sep 15 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,
2 + 3 x + x^2,
4 + 8 x + 5 x^2 + x^3.
First 5 rows of array:
 1    1
 2    3    1
 4    8    5   1
12   28   23   8   1
48  124  120  55  12  1
		

Crossrefs

Cf. A000045, A232896 (T(n,n)), A374848, A375048, A375049.

Programs

Mathematica

s[n_] := x + 1; 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 *)