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%I #12 Jun 07 2021 14:54:46
%S 0,1,0,0,1,0,1,1,1,0,1,2,2,1,0,2,1,3,3,1,0,1,2,4,4,4,1,0,2,2,5,27,5,5,
%T 1,0,0,3,6,28,256,6,6,1,0,1,1,7,30,257,3125,7,7,1,0,0,2,8,31,260,3126,
%U 46656,8,8,1,0,1,2,9,81,261,3130,46657,823543,9,9,1,0
%N T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Goodstein's_theorem#Hereditary_base-n_notation">Hereditary base-n notation</a>
%F T(n, n) = A343255(n).
%F T(n, 0) = A345021(n).
%F T(n, 1) = A000120(n).
%F T(n, 2) = n.
%F T(n, 3) = A222112(n-1).
%F T(0, k) = 0.
%F T(1, k) = 1.
%F T(2, k) = k.
%F T(3, k) = k + 1.
%F T(4, k) = k^k = A000312(k).
%F T(5, k) = k^k + 1 = A014566(k).
%F T(6, k) = k^k + k = A066068(k).
%F T(7, k) = k^k + k + 1 = A066279(k).
%F T(16, k) = k^k^k = A002488(k).
%F T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).
%e Array T(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9
%e ---+-------------------------------------------------------------------
%e 0| 0 0 0 0 0 0 0 0 0 0
%e 1| 1 1 1 1 1 1 1 1 1 1
%e 2| 0 1 2 3 4 5 6 7 8 9
%e 3| 1 2 3 4 5 6 7 8 9 10
%e 4| 1 1 4 27 256 3125 46656 823543 16777216 387420489
%e 5| 2 2 5 28 257 3126 46657 823544 16777217 387420490
%e 6| 1 2 6 30 260 3130 46662 823550 16777224 387420498
%e 7| 2 3 7 31 261 3131 46663 823551 16777225 387420499
%e 8| 0 1 8 81 1024 15625 279936 5764801 134217728 3486784401
%e 9| 1 2 9 82 1025 15626 279937 5764802 134217729 3486784402
%e 10| 0 2 10 84 1028 15630 279942 5764808 134217736 3486784410
%o (PARI) T(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n,2); v+=k^T(e,k)); v }
%Y See A341907 for a similar sequence.
%Y Cf. A000120, A000312, A002488, A014566, A066068, A066279, A222112, A343255, A345021.
%K nonn,tabl,base
%O 0,12
%A _Rémy Sigrist_, Jun 04 2021
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