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%I #28 Jul 14 2022 22:55:20
%S 0,1,1,2,2,1,3,3,4,3,4,4,7,8,5,5,5,10,13,16,11,6,6,13,18,27,32,21,7,7,
%T 16,23,38,53,64,43,8,8,19,28,49,74,107,128,85,9,9,22,33,60,95,150,213,
%U 256,171,10,10,25,38,71,116,193,298,427,512,341
%N Array read by upwards antidiagonals T(n,k) = J(k) + n*J(k+1) where J(n) = A001045(n) is the Jacobsthal numbers.
%F T(n, k) = (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3.
%F G.f.: (x*(y-1) - y)/((x - 1)^2*(y + 1)*(2*y - 1)). - _Stefano Spezia_, Jul 13 2022
%e Row n=0 is A001045(k), then for further rows we successively add A001045(k+1).
%e k=0 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10
%e n=0: 0 1 1 3 5 11 21 43 85 171 ... = A001045
%e n=1: 1 2 4 8 16 32 64 128 256 512 ... = A000079
%e n=2: 2 3 7 13 27 53 107 213 427 853 ... = A048573
%e n=3: 3 4 10 18 38 74 150 298 598 1194 ... = A171160
%e n=4: 4 5 13 23 49 95 193 383 769 1535 ... = abs(A140683)
%e ...
%t T[n_, k_] := (2^k - (-1)^k + n*(2^(k + 1) + (-1)^k))/3; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 13 2022 *)
%Y Cf. A001477, A000027, A016777, A016885, A017449, A321373.
%Y Antidiagonal sums give A320933(n+1).
%K nonn,tabl,easy
%O 0,4
%A _Paul Curtz_, Jul 13 2022
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