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%I #9 May 25 2022 10:49:45
%S 0,0,1,0,1,0,0,1,1,-1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,1,2,-1,0,0,0,0,1,2,
%T 2,0,0,0,0,0,0,1,3,4,0,0,0,0,0,0,1,2,6,5,0,0,0,0,0,0,0,1,4,10,8,0,0,0,
%U 0,0,0,0,1,2,7,18,11,0,0,0,0,0,0,0,0,1,4,14,31,18,0,0
%N Array read by ascending antidiagonals. Inverse Euler transform of the right-shifted k-bonacci numbers.
%e Array starts:
%e [0] 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e [1] 0, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...
%e [2] 0, 1, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, ...
%e [3] 0, 0, 1, 1, 2, 3, 6, 10, 18, 31, 56, 96, 172, ...
%e [4] 0, 0, 0, 1, 1, 2, 4, 7, 14, 26, 50, 93, 178, ...
%e [5] 0, 0, 0, 0, 1, 1, 2, 4, 8, 15, 30, 58, 114, ...
%e [6] 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 62, ...
%e [7] 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, ...
%e [8] 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, ...
%e [9] 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, ...
%e .
%e Compare the rows with the columns of A349802.
%p read transforms;
%p F := proc(n, k) option remember;
%p ifelse(k < 2, k, add(F(n, k-j), j = 1..min(n, k))) end:
%p Frow := (n, len) -> [seq(0, j = 0..n-3), seq(F(n, k), k = 0..len)]:
%p Arow := (n, len) -> EULERi(Frow(n, len)):
%p for n from 0 to 9 do Arow(n, 14 - n) od;
%Y Rows are the inverse Euler transforms of A063524, A057427, A000045, A000073, A000078, A001591, A001592.
%Y Cf. A092921, A349802
%K sign,tabl
%O 0,26
%A _Peter Luschny_, Dec 05 2021
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