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%I #31 Nov 28 2023 14:27:32
%S 1,0,2,-1,1,4,-2,0,3,8,-3,-1,2,7,16,-4,-2,1,6,15,32,-5,-3,0,5,14,31,
%T 64,-6,-4,-1,4,13,30,63,128,-7,-5,-2,3,12,29,62,127,256,-8,-6,-3,2,11,
%U 28,61,126,255,512,-9,-7,-4,1,10,27,60,125,254,511,1024
%N Square array T(n, k) = 2^k - n, read by ascending antidiagonals.
%H Paolo Xausa, <a href="/A367559/b367559.txt">Table of n, a(n) for n = 0..11475</a> (antidiagonals 0..150, flattened).
%F G.f. of row n: 1/(1-2*x) - n/(1-x).
%F E.g.f. of row n: exp(2*x) - n*exp(x).
%F T(0, k) = 2^k = A000079(k).
%F T(1, k) = 2^k - 1 = A000225(k).
%F T(2, k) = 2^k - 2 = A000918(k).
%F T(3, k) = 2^k - 3 = A036563(k).
%F T(5, k) = 2^k - 5 = A168616(k).
%F T(9, k) = 2^k - 9 = A185346(k).
%F T(10, k) = 2^k - 10 = A246168(k).
%F T(n, k) = 3*T(n, k-1) - 2*T(n, k-2) for k > 1.
%F T(n+1, k) = T(n, k) + 1.
%F T(n, n) = 2^n - n = A000325(n).
%F Sum_{k = 0..n} T(n - k, k) = A084634(n).
%F a(n) = 2^A002262(n) - A025581(n).
%F G.f.: (1 - 2*x - y + 3*x*y)/((1 - x)^2*(1 - y)*(1 - 2*y)). - _Stefano Spezia_, Nov 27 2023
%e This sequence as square array T(n, k):
%e n\k 0 1 2 3 4 5 6 7 8 9 10.
%e ---------------------------------------------------------.
%e 0 : 1 2 4 8 16 32 64 128 256 512 1024.
%e 1 : 0 1 3 7 15 31 63 127 255 511 1023.
%e 2 : -1 0 2 6 14 30 62 126 254 510 1022.
%e 3 : -2 -1 1 5 13 29 61 125 253 509 1021.
%e 4 : -3 -2 0 4 12 28 60 124 252 508 1020.
%e 5 : -4 -3 -1 3 11 27 59 123 251 507 1019.
%e 6 : -5 -4 -2 2 10 26 58 122 250 506 1018.
%e 7 : -6 -5 -3 1 9 25 57 121 249 505 1017.
%e 8 : -7 -6 -4 0 8 24 56 120 248 504 1016.
%e 9 : -8 -7 -5 -1 7 23 55 119 247 503 1015.
%e 10: -9 -8 -6 -2 6 22 54 118 246 502 1014.
%t Table[2^k-n+k,{n,0,10},{k,0,n}] (* _Paolo Xausa_, Nov 28 2023 *)
%o (PARI) T(n, k) = 2^k-n \\ _Thomas Scheuerle_, Nov 23 2023
%Y Cf. A000079, A000225, A000325, A000295.
%Y Cf. A000325, A000918, A084634, A036563.
%Y Cf. A168616, A185346, A246168.
%Y Cf. A002262, A025581.
%K easy,sign,tabl
%O 0,3
%A _Paul Curtz_, Nov 22 2023
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