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A375043 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+2. See Comments.

%I A375043 #4 Sep 20 2024 06:08:08
%S A375043 2,1,4,6,2,8,32,34,10,16,144,388,360,100,32,560,3224,7316,6320,1700,
%T A375043 64,1952,21008,98456,202856,167720,44200,128,6272,114240,974208,
%U A375043 4048584,7841112,6294040,1635400,256,18944,542080,7660416,56807568,218111424,404643680
%N A375043 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+2. See Comments.
%C A375043 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.
%e A375043 First 3 polynomials in s(x)**t(x) are
%e A375043 2 + x,
%e A375043 4 + 6 x + 2 x^2,
%e A375043 8 + 32 x + 34 x^2 + 10 x^3.
%e A375043 First 5 rows of array:
%e A375043  2    1
%e A375043  4    6     2
%e A375043  8   32    34    10
%e A375043 16  144   388   360   100
%e A375043 32  560  3224  7316  6320  1700
%t A375043 s[n_] := n^2  x; t[n_] := x + 2;
%t A375043 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375043 Table[Expand[u[n]], {n, 0, 10}]
%t A375043 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375043 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)
%Y A375043 Cf. A000290, A101686 (T(n,n+1)), A374848, A375041, A375042.
%K A375043 nonn,tabf
%O A375043 1,1
%A A375043 _Clark Kimberling_, Sep 14 2024