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COMMENTS
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Hankel transform of 1,1,3,11,45,... (see A026375). Binomial transform of A015448.
A production matrix for the sequence is M =
1, 1, 0, 0, 0, ...
1, 0, 5, 0, 0, ...
1, 0, 0, 5, 0, ...
1, 0, 0, 0, 5, ...
...
Take powers of M, extracting the upper left terms; getting
the sequence starting (1, 1, 2, 8, 40, 208, ...). (End)
The sequence is N=5 in an infinite set of INVERT transforms of powers of N prefaced with a "1". (1, 2, 8, 40, ...) is the INVERT transform of (1, 1, 5, 25, 125, ...). The first six of such sequences are shown in A006012 (N=3). - Gary W. Adamson, Jul 24 2016
The sequence is the first in an infinite set in which we perform the operation for matrix M (Cf. Jul 22 2016), but change the left border successively from (1, 1, 1, 1, ...) then to (1, 2, 2, 2, ...), then (1, 3, 3, 3, ...) ...; generally (1, N, N, N, ...). Extracting the upper left terms of each matrix operation, we obtain the infinite set beginning:
N=1 (A154626): 1, 2, 8, 40, 208, 1088, ...
N=2 (A084120): 1, 3, 15, 81, 441, 1403, ...
N=3 (A180034): 1, 4, 22, 124, 700, 3952, ...
N=4 (A001653): 1, 5, 29, 169, 985, 5741, ...
N=5 (A000400): 1, 6, 36, 216, 1296, 7776, ...
N=6 (A015451): 1, 7, 43, 265, 1633, 10063, ...
N=7 (A180029): 1, 8, 50, 316, 1996, 12608, ...
N=8 (A180028): 1, 9, 57, 369, 1285, 15417, ...
N=9 (.......): 1, 10, 64, 424, 2800, 18496, ...
N=10 (A123361): 1, 11, 71, 481, 3241, 21851, ...
N=11 (.......): 1, 12, 78, 540, 3708, 25488, ...
... Each of the sequences begins (1, (N+1), (7*N + 1),
(40*N + (N-1)^2), ... (End)
The set of infinite sequences shown (Cf. comment of Jul 27 2016), can be generated from the matrices P = [(1,N; 1,5]^n, (N=1,2,3,...) by extracting the upper left terms. Example: N=6 sequence (A015451): (1, 7, 43, 265, ...) can be generated from the matrix P = [(1,6); (1,5)]^n. - Gary W. Adamson, Jul 28 2016
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