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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.
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%I #5 Sep 20 2024 06:08:36

%S 2,2,6,10,4,30,62,40,8,270,618,484,152,16,4590,11046,9464,3552,576,32,

%T 151470,373698,334404,136144,26112,2208,64,9845550,24593310,22483656,

%U 9518168,1969568,195744,8576,128,1270075950,3192228090,2949578244,1272810984

%N 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 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

%e First 3 polynomials in s(x)**t(x) are

%e 2 + 2x,

%e 6 + 10 x + 4 x^2,

%e 30 + 62 x + 40 x^2 + 8 x^3.

%e First 5 rows of array:

%e 2 2

%e 6 10 4

%e 30 62 40 8

%e 270 618 484 152 16

%e 4590 11046 9464 3552 576 32

%t s[n_] := 2^n x; t[n_] := 2x + 1;

%t u[n_] := Product[s[k] + t[n - k], {k, 0, n}]

%t Table[Expand[u[n]], {n, 0, 10}]

%t Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]] (* array *)

%t Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]] (* sequence *)

%Y Cf. A000079, A028361 (column 2), A000079 (T(n,n+1)), A374848.

%K nonn,tabf

%O 1,1

%A _Clark Kimberling_, Sep 15 2024