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A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

%I A385178 #76 Jun 28 2025 17:56:20
%S A385178 0,1,1,3,4,5,7,10,14,19,15,22,32,46,65,31,46,68,100,146,211,63,94,140,
%T A385178 208,308,454,665,127,190,284,424,632,940,1394,2059,255,382,572,856,
%U A385178 1280,1912,2852,4246,6305,511,766,1148,1720,2576,3856,5768,8620,12866,19171
%N A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.
%F A385178 T(n,n) = 3^n - 2^n = A001047(n).
%F A385178 T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
%F A385178 T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
%F A385178 T(2n,n) = A248216(n+1).
%e A385178 Triangle begins:
%e A385178     0;
%e A385178     1,   1;
%e A385178     3,   4,    5;
%e A385178     7,  10,   14,   19;
%e A385178    15,  22,   32,   46,   65;
%e A385178    31,  46,   68,  100,  146,  211;
%e A385178    63,  94,  140,  208,  308,  454,  665;
%e A385178   127, 190,  284,  424,  632,  940, 1394, 2059;
%e A385178   255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
%e A385178   511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
%e A385178   ...
%p A385178 T:= proc(n,k) option remember;
%p A385178      `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
%p A385178     end:
%p A385178 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jun 24 2025
%t A385178 t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 20 2025 *)
%o A385178 (Magma) /* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // _Vincenzo Librandi_, Jun 27 2025
%Y A385178 Columns k=0..2: A000225, A033484, A053209 (sans 1).
%Y A385178 Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
%Y A385178 Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.
%K A385178 nonn,tabl,easy
%O A385178 0,4
%A A385178 _Paul Curtz_, Jun 20 2025