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A349903 Array read by ascending antidiagonals. Inverse Euler transform of the right-shifted k-bonacci numbers.

%I A349903 #9 May 25 2022 10:49:45
%S A349903 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 A349903 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 A349903 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 A349903 Array read by ascending antidiagonals. Inverse Euler transform of the right-shifted k-bonacci numbers.
%e A349903 Array starts:
%e A349903 [0] 0, 1, 0, -1, 0,  0, 0,  0,  0,  0,  0,  0,   0, ...
%e A349903 [1] 0, 1, 1,  0, 0, -1, 0,  0,  0,  0,  0,  0,   0, ...
%e A349903 [2] 0, 1, 1,  1, 2,  2, 4,  5,  8, 11, 18, 25,  40, ...
%e A349903 [3] 0, 0, 1,  1, 2,  3, 6, 10, 18, 31, 56, 96, 172, ...
%e A349903 [4] 0, 0, 0,  1, 1,  2, 4,  7, 14, 26, 50, 93, 178, ...
%e A349903 [5] 0, 0, 0,  0, 1,  1, 2,  4,  8, 15, 30, 58, 114, ...
%e A349903 [6] 0, 0, 0,  0, 0,  1, 1,  2,  4,  8, 16, 31,  62, ...
%e A349903 [7] 0, 0, 0,  0, 0,  0, 1,  1,  2,  4,  8, 16,  32, ...
%e A349903 [8] 0, 0, 0,  0, 0,  0, 0,  1,  1,  2,  4,  8,  16, ...
%e A349903 [9] 0, 0, 0,  0, 0,  0, 0,  0,  1,  1,  2,  4,   8, ...
%e A349903 .
%e A349903 Compare the rows with the columns of A349802.
%p A349903 read transforms;
%p A349903 F := proc(n, k) option remember;
%p A349903      ifelse(k < 2, k, add(F(n, k-j), j = 1..min(n, k))) end:
%p A349903 Frow := (n, len) -> [seq(0, j = 0..n-3), seq(F(n, k), k = 0..len)]:
%p A349903 Arow := (n, len) -> EULERi(Frow(n, len)):
%p A349903 for n from 0 to 9 do Arow(n, 14 - n) od;
%Y A349903 Rows are the inverse Euler transforms of A063524, A057427, A000045, A000073, A000078, A001591, A001592.
%Y A349903 Cf. A092921, A349802
%K A349903 sign,tabl
%O A349903 0,26
%A A349903 _Peter Luschny_, Dec 05 2021