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%I #22 May 13 2023 15:33:21
%S 1,1,1,2,6,27,180,1786,26094,559127,17535396,804131875,53833201737
%N Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.
%C From _T. D. Noe_, Feb 05 2007: (Start)
%C The slowest-growing sequence of length n is 1,2,4,6,...,2(n-1). The fastest-growing sequence is 1,2,4,8,...,2^(n-1).
%C The ratio a(n+1)a(n-1)/a(n)^2 appears to converge to a constant near 1.46, which is the approximate growth rate of A001609. Are the sequences related?
%C (End)
%C Also, a(n) is the number of (not necessarily increasing) positive integer sequences of length n-1 with Gilbreath transform (1, ..., 1). - _Pontus von Brömssen_, May 13 2023
%e The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
%e 1
%e 2 1
%e 4 2 1
%e 6 2 0 1
%e 10 4 2 2 1
%Y Cf. A001609, A036262, A363002, A363003, A363004, A363005.
%Y Cf. also A136465, the total number of increasing sequences with the same maximum length. [From _Charles R Greathouse IV_, Aug 08 2010]
%K nonn,more
%O 1,4
%A _John W. Layman_, Mar 28 2003
%E More terms from _T. D. Noe_, Feb 05 2007
%E Added "positive" to definition. - _N. J. A. Sloane_, May 13 2023
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