A080839 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,...}.

1, 1, 1, 2, 6, 27, 180, 1786, 26094, 559127, 17535396, 804131875, 53833201737

Offset: 1

Author

John W. Layman, Mar 28 2003

Keywords

• nonn
• more

Comments

From T. D. Noe, Feb 05 2007: (Start)

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).

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?

(End)

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

Examples

			The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
   1
   2 1
   4 2 1
   6 2 0 1
  10 4 2 2 1
		

Crossrefs

Cf. A001609, A036262, A363002, A363003, A363004, A363005.

Cf. also A136465, the total number of increasing sequences with the same maximum length. [From Charles R Greathouse IV, Aug 08 2010]

Extensions

More terms from T. D. Noe, Feb 05 2007

Added "positive" to definition. - N. J. A. Sloane, May 13 2023