cp's OEIS Frontend

A375042 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) = 2x+1. See Comments.

%I A375042 #7 Sep 13 2024 14:04:06
%S A375042 1,2,1,5,6,1,11,36,36,1,22,157,432,396,1,40,553,3258,8172,7128,1,67,
%T A375042 1633,18189,96138,227772,192456,1,105,4179,80243,787320,3881016,
%U A375042 8847792,7313328,1,156,9534,293372,4879713,44034336,206779608,458550720,372979728,1
%N A375042 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) = 2x+1. See Comments.
%C A375042 See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.
%e A375042 First 3 polynomials in s(x)**t(x) are
%e A375042   1 + 2x,
%e A375042   1 + 5 x + 6 x^2,
%e A375042   1 + 11 x + 36 x^2 + 36 x^3.
%e A375042 First 5 rows of array:
%e A375042   1   2
%e A375042   1   5      6
%e A375042   1   11    36    36
%e A375042   1   22   157   432   396
%e A375042   1   40   553  3258  8172  7128
%t A375042 s[n_] := n^2  x; t[n_] := 1 + 2 x;
%t A375042 u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
%t A375042 Table[Expand[u[n]], {n, 0, 10}]
%t A375042 Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
%t A375042 Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]    (* sequence *)
%Y A375042 Cf. A000290, A277355 ((1/2)T(n,n+1)), A374848, A375041, A375043.
%K A375042 nonn,tabf
%O A375042 1,2
%A A375042 _Clark Kimberling_, Sep 11 2024