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A363003 Number of integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

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%I A363003 #9 May 13 2023 13:48:09
%S A363003 1,1,2,6,26,166,1562,21614,438594,13032614,566069882
%N A363003 Number of integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).
%C A363003 a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).
%C A363003 Negative terms are permitted.
%e A363003 For n = 4, the following 13 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 26.
%e A363003   (1, 2, 0, -4),
%e A363003   (1, 2, 0, -2),
%e A363003   (1, 2, 0,  0),
%e A363003   (1, 2, 0,  2),
%e A363003   (1, 2, 0,  4),
%e A363003   (1, 2, 2,  0),
%e A363003   (1, 2, 2,  2),
%e A363003   (1, 2, 2,  4),
%e A363003   (1, 2, 4,  0),
%e A363003   (1, 2, 4,  2),
%e A363003   (1, 2, 4,  4),
%e A363003   (1, 2, 4,  6),
%e A363003   (1, 2, 4,  8).
%Y A363003 Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363004 (distinct positive integers), A363005 (distinct integers).
%K A363003 nonn,more
%O A363003 0,3
%A A363003 _Pontus von Brömssen_, May 13 2023