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A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

%I A356036 #17 Aug 05 2022 13:15:39
%S A356036 1,3,4,9,12,16,27,36,48,64,81,108,144,192,256,243,324,432,576,768,
%T A356036 1024,729,972,1296,1728,2304,3072,4096,2187,2916,3888,5184,6912,9216,
%U A356036 12288,16384,6561,8748,11664,15552,20736,27648,36864,49152,65536,19683,26244,34992,46656,62208,82944,110592,147456,196608,262144
%N A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.
%C A356036 This is Boethius's triangle, with rows read as columns. See the link and reference.
%D A356036 Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.
%H A356036 Anicius Manlius Severinus Boethius, <a href="https://archive.org/details/ita-bnc-in2-00001756-001/page/n54/mode/1up?view=theater">De Institutione Arithmetica</a>, 1488, p. 55 of 104, top of left column.
%F A356036 T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.
%F A356036 G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).
%e A356036 The triangle T begins:
%e A356036 n\k     0     1      2      3      4      5      6      7      8      9  ...
%e A356036 0:      1
%e A356036 1:      3     4
%e A356036 2:      9    12     16
%e A356036 3:     27    36     48     64
%e A356036 4:     81   108    144    192    256
%e A356036 5:    243   324    432    576    768   1024
%e A356036 6:    729   972   1296   1728   2304   3072   4096
%e A356036 7:   2187  2916   3888   5184   6912   9216  12288  16384
%e A356036 8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
%e A356036 9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
%e A356036 ...
%t A356036 T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Aug 05 2022 *)
%Y A356036 Columns: A000244, A003946, A257970, ...
%Y A356036 Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
%Y A356036 Row sums: A005061(n+1).
%K A356036 nonn,tabl,easy
%O A356036 0,2
%A A356036 _Wolfdieter Lang_, Aug 01 2022