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A375041 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+1. See Comments.

%I A375041 #8 Sep 13 2024 14:03:39
%S A375041 1,1,1,3,2,1,8,17,10,1,18,97,180,100,1,35,403,1829,3160,1700,1,61,
%T A375041 1313,12307,50714,83860,44200,1,98,3570,60888,506073,1960278,3147020,
%U A375041 1635400,1,148,8470,239388,3550473,27263928,101160920,158986400,81770000,1,213
%N A375041 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+1. See Comments.
%C A375041 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.
%e A375041 First 3 polynomials in s(x)**t(x) are
%e A375041   1 + x,
%e A375041   1 + 3 x + 2 x^2,
%e A375041   1 + 8 x + 17 x^2 + 10 x^3.
%e A375041 First 5 rows of array:
%e A375041   1    1
%e A375041   1    3     2
%e A375041   1    8    17    10
%e A375041   1   18    97   180   100
%e A375041   1   35  4034  1829  3160  1700
%t A375041 s[n_] := n^2  x; t[n_] := 1 + x;
%t A375041 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375041 Table[Expand[u[n]], {n, 0, 10}]
%t A375041 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375041 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)
%Y A375041 Cf. A000290, A081489 (column 2), A101686 (T(n,n+1)), A374848, A375042, A375043.
%K A375041 nonn,tabf
%O A375041 1,4
%A A375041 _Clark Kimberling_, Sep 11 2024