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A375045 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 and t(x) = 2x+1. See Comments.

%I A375045 #5 Sep 20 2024 06:08:36
%S A375045 2,2,6,10,4,30,62,40,8,270,618,484,152,16,4590,11046,9464,3552,576,32,
%T A375045 151470,373698,334404,136144,26112,2208,64,9845550,24593310,22483656,
%U A375045 9518168,1969568,195744,8576,128,1270075950,3192228090,2949578244,1272810984
%N A375045 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 and t(x) = 2x+1. See Comments.
%C A375045 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.
%e A375045 First 3 polynomials in s(x)**t(x) are
%e A375045 2 + 2x,
%e A375045 6 + 10 x + 4 x^2,
%e A375045 30 + 62 x + 40 x^2 + 8 x^3.
%e A375045 First 5 rows of array:
%e A375045    2      2
%e A375045    6     10     4
%e A375045   30     62    40     8
%e A375045  270    618   484   152   16
%e A375045 4590  11046  9464  3552  576  32
%t A375045 s[n_] := 2^n  x; t[n_] := 2x + 1;
%t A375045 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375045 Table[Expand[u[n]], {n, 0, 10}]
%t A375045 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375045 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)
%Y A375045 Cf. A000079, A028361 (column 2), A000079 (T(n,n+1)), A374848.
%K A375045 nonn,tabf
%O A375045 1,1
%A A375045 _Clark Kimberling_, Sep 15 2024