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

1, 2, 8, 40, 208, 1088, 5696, 29824, 156160, 817664, 4281344, 22417408, 117379072, 614604800, 3218112512, 16850255872, 88229085184, 461973487616, 2418924584960, 12665653559296, 66318223015936, 347246723858432, 1818207451086848, 9520257811087360

Offset: 0

Author

Paul Barry, Jan 13 2009

Keywords

• easy
• nonn
• changed

Comments

Hankel transform of 1,1,3,11,45,... (see A026375). Binomial transform of A015448.

From Gary W. Adamson, Jul 22 2016: (Start)

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

From Gary W. Adamson, Jul 27 2016: (Start)

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

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 0..1000
• Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
• Index entries for linear recurrences with constant coefficients, signature (6,-4).

Crossrefs

Cf. A006012, A084120, A180034, A001653, A000400, A015451, A180029, A180028, A123362.

Programs

Magma

[n le 2 select (n) else 6*Self(n-1)-4*Self(n-2): n in [1..25]]; // Vincenzo Librandi, May 15 2015

Mathematica

LinearRecurrence[{6, -4}, {1, 2}, 30] (* Vincenzo Librandi, May 15 2015 *)

PARI

Vec((1-4*x) / (1-6*x+4*x^2) + O(x^30)) \\ Colin Barker, Sep 22 2017

Formula

G.f.: (1 - 4*x) / (1 - 6*x + 4*x^2).

a(n) = A084326(n+1) - 4*A084326(n). - R. J. Mathar, Jul 19 2012

From Colin Barker, Sep 22 2017: (Start)

a(n) = (((3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n)) / (2*sqrt(5)).

a(n) = 6*a(n-1) - 4*a(n-2) for n>1. (End)

E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) - sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Aug 26 2025