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A154626 a(n) = 2^n*A001519(n).

%I A154626 #54 Aug 26 2025 14:25:25
%S A154626 1,2,8,40,208,1088,5696,29824,156160,817664,4281344,22417408,
%T A154626 117379072,614604800,3218112512,16850255872,88229085184,461973487616,
%U A154626 2418924584960,12665653559296,66318223015936,347246723858432,1818207451086848,9520257811087360
%N A154626 a(n) = 2^n*A001519(n).
%C A154626 Hankel transform of 1,1,3,11,45,... (see A026375). Binomial transform of A015448.
%C A154626 From _Gary W. Adamson_, Jul 22 2016: (Start)
%C A154626 A production matrix for the sequence is M =
%C A154626   1, 1, 0, 0, 0, ...
%C A154626   1, 0, 5, 0, 0, ...
%C A154626   1, 0, 0, 5, 0, ...
%C A154626   1, 0, 0, 0, 5, ...
%C A154626   ...
%C A154626 Take powers of M, extracting the upper left terms; getting
%C A154626 the sequence starting (1, 1, 2, 8, 40, 208, ...). (End)
%C A154626 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
%C A154626 From _Gary W. Adamson_, Jul 27 2016: (Start)
%C A154626 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:
%C A154626   N=1  (A154626): 1,  2,  8,  40,  208,  1088, ...
%C A154626   N=2  (A084120): 1,  3, 15,  81,  441,  1403, ...
%C A154626   N=3  (A180034): 1,  4, 22, 124,  700,  3952, ...
%C A154626   N=4  (A001653): 1,  5, 29, 169,  985,  5741, ...
%C A154626   N=5  (A000400): 1,  6, 36, 216, 1296,  7776, ...
%C A154626   N=6  (A015451): 1,  7, 43, 265, 1633, 10063, ...
%C A154626   N=7  (A180029): 1,  8, 50, 316, 1996, 12608, ...
%C A154626   N=8  (A180028): 1,  9, 57, 369, 1285, 15417, ...
%C A154626   N=9  (.......): 1, 10, 64, 424, 2800, 18496, ...
%C A154626   N=10 (A123361): 1, 11, 71, 481, 3241, 21851, ...
%C A154626   N=11 (.......): 1, 12, 78, 540, 3708, 25488, ...
%C A154626 ... Each of the sequences begins (1, (N+1), (7*N + 1),
%C A154626   (40*N + (N-1)^2), ... (End)
%C A154626 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
%H A154626 Karl V. Keller, Jr., <a href="/A154626/b154626.txt">Table of n, a(n) for n = 0..1000</a>
%H A154626 Bishal Deb, <a href="https://arxiv.org/abs/2508.13709">Cyclic sieving phenomena via combinatorics of continued fractions</a>, arXiv:2508.13709 [math.CO], 2025. See p. 42.
%H A154626 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).
%F A154626 G.f.: (1 - 4*x) / (1 - 6*x + 4*x^2).
%F A154626 a(n) = A084326(n+1) - 4*A084326(n). - _R. J. Mathar_, Jul 19 2012
%F A154626 From _Colin Barker_, Sep 22 2017: (Start)
%F A154626 a(n) = (((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n)) / (2*sqrt(5)).
%F A154626 a(n) = 6*a(n-1) - 4*a(n-2) for n>1. (End)
%F A154626 E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) - sqrt(5)*sinh(sqrt(5)*x))/5. - _Stefano Spezia_, Aug 26 2025
%t A154626 LinearRecurrence[{6, -4}, {1, 2}, 30] (* _Vincenzo Librandi_, May 15 2015 *)
%o A154626 (Magma) [n le 2 select (n) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, May 15 2015
%o A154626 (PARI) Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ _Colin Barker_, Sep 22 2017
%Y A154626 Cf. A006012, A084120, A180034, A001653, A000400, A015451, A180029, A180028, A123362.
%K A154626 easy,nonn,changed
%O A154626 0,2
%A A154626 _Paul Barry_, Jan 13 2009