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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.
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%I #17 Aug 05 2022 13:15:39

%S 1,3,4,9,12,16,27,36,48,64,81,108,144,192,256,243,324,432,576,768,

%T 1024,729,972,1296,1728,2304,3072,4096,2187,2916,3888,5184,6912,9216,

%U 12288,16384,6561,8748,11664,15552,20736,27648,36864,49152,65536,19683,26244,34992,46656,62208,82944,110592,147456,196608,262144

%N 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 This is Boethius's triangle, with rows read as columns. See the link and reference.

%D Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

%H 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 T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

%F 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 The triangle T begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: 3 4

%e 2: 9 12 16

%e 3: 27 36 48 64

%e 4: 81 108 144 192 256

%e 5: 243 324 432 576 768 1024

%e 6: 729 972 1296 1728 2304 3072 4096

%e 7: 2187 2916 3888 5184 6912 9216 12288 16384

%e 8: 6561 8748 11664 15552 20736 27648 36864 49152 65536

%e 9: 19683 26244 34992 46656 62208 82944 110592 147456 196608 262144

%e ...

%t 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 Columns: A000244, A003946, A257970, ...

%Y Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...

%Y Row sums: A005061(n+1).

%K nonn,tabl,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 01 2022