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A375044 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) = 2^n x and t(x) = x+1. See Comments.

%I A375044 #4 Sep 20 2024 06:08:24
%S A375044 1,2,1,5,6,1,10,31,30,1,19,121,309,270,1,36,444,2366,5523,4590,1,69,
%T A375044 1632,17018,83601,186849,151470,1,134,6117,123098,1189771,5620914,
%U A375044 12296655,9845550,1,263,23403,912191,17069413,159101373,737394561,1596114045
%N A375044 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) = 2^n x and t(x) = x+1. See Comments.
%C A375044 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.
%e A375044 First 3 polynomials in s(x)**t(x) are
%e A375044 1 + 2x,
%e A375044 1 + 5 x + 6 x^2,
%e A375044 1 + 10 x + 31 x^2 + 30 x^3.
%e A375044 First 5 rows of array:
%e A375044 1    2
%e A375044 1    5     6
%e A375044 1   10    31    30
%e A375044 1   19   121   309    270
%e A375044 1   36   444  2366   5523   4590
%t A375044 s[n_] := 2^n  x; t[n_] := x + 1;
%t A375044 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375044 Table[Expand[u[n]], {n, 0, 10}]
%t A375044 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375044 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)
%Y A375044 Cf. A000290, A028361 (T(n,n+1)), A374848.
%K A375044 nonn,tabf
%O A375044 1,2
%A A375044 _Clark Kimberling_, Sep 15 2024