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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.

%I A375049 #8 Aug 02 2024 18:59:58
%S A375049 0,2,1,4,4,2,10,16,8,16,64,96,64,16,162,594,864,624,224,32,3600,11280,
%T A375049 14596,9984,3808,768,64,147456,393216,443392,273920,100096,21632,2560,
%U A375049 128,12320100,27335880,26086356,13971408,4589488,946176,119488,8448,256
%N 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.
%C A375049 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.
%e A375049 First 3 polynomials in s(x)**t(x) are
%e A375049   0 + 2x,
%e A375049   1 + 4 x + 4x^2,
%e A375049   2 + 10 x + 16 x^2 + 8 x^3.
%e A375049 First 5 rows of array:
%e A375049   0   2
%e A375049   1   4   4
%e A375049   2   10  16   8
%e A375049   16  64  96  64  16
%e A375049   162 594 864 624 224 32
%t A375049 s[n_] := x + Fibonacci[n]; t[n_] := Fibonacci[n];
%t A375049 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375049 Table[Expand[u[n]], {n, 0, 10}]
%t A375049 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375049 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]    (* sequence *)
%Y A375049 Cf. A000045, A000079 (T(n,n+1)), A374848, A375047, A375048.
%K A375049 nonn,tabf
%O A375049 1,2
%A A375049 _Clark Kimberling_, Jul 31 2024