A104538 -id:A104538 - cp's OEIS frontend

A120580 Hankel transform of Sum_{k=0..n} C(2k,k).

1, 0, -4, -8, 0, 32, 64, 0, -256, -512, 0, 2048, 4096, 0, -16384, -32768, 0, 131072, 262144, 0, -1048576, -2097152, 0, 8388608, 16777216, 0, -67108864, -134217728, 0, 536870912, 1073741824, 0, -4294967296, -8589934592, 0, 34359738368, 68719476736, 0, -274877906944, -549755813888, 0

Offset: 0

Paul Barry, Jun 15 2006

Hankel transform of A006134.
Hankel transform of A098479. - Paul Barry, Sep 19 2008
Hankel transform of A025565. - Paul Barry, Mar 26 2010

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A006134, A025565, A098479, A104538.

LinearRecurrence[{2,-4},{1,0},50] (* Harvey P. Dale, Feb 13 2023 *)

G.f.: (1-2x)/(1-2x+4x^2).
a(n) = 2^n*(cos(Pi*n/3)-sin(Pi*n/3)/sqrt(3)).
E.g.f.: exp(x)*(cos(sqrt(3)*x) - sin(sqrt(3)*x)/sqrt(3)). - Stefano Spezia, Jul 15 2024

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Original entry on oeis.org

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 19 2017

This is the inverse binomial transform of A157241.
Successive differences of A157241 begin:
0,   1,   3,   3,   -5,  -21,  -21,    43,   171,   171, ... = A157241
1,   2,   0,  -8,  -16,    0,   64,   128,     0,  -512, ... = A088138
1,  -2,  -8,  -8,   16,   64,   64,  -128,  -512,  -512, ... = A138230
-3, -6,   0,  24,   48,    0, -192,  -384,     0,  1536, ...
-3,  6,  24,  24,  -48, -192, -192,   384,  1536,  1536, ...
9,  18,   0, -72, -144,    0,  576,  1152,     0, -4608, ...
9, -18, -72  -72,  144,  576,  576, -1152, -4608, -4608, ...
...
a(n) is the n-th term of the first column.
Successive differences of a(n) begin:
0,     1,    1,   -3,   -3,    9,     9,   -27,   -27,    81, ...
1,     0,   -4,    0,   12,    0,   -36,     0,   108,     0, ...
-1,   -4,    4,   12,  -12,  -36,    36,   108,  -108,  -324, ...
-3,    8,    8,  -24,  -24,   72,    72,  -216,  -216,   648, ...
11,    0,  -32,    0,   96,    0,  -288,     0,   864,     0, ...
-11, -32,   32,   96,  -96, -288,   288,   864,  -864, -2592, ...
-21,  64,   64, -192, -192,  576,   576, -1728, -1728,  5184, ...
85,    0, -256,    0,  768,    0, -2304,     0,  6912,     0, ...
...
First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

Index entries for linear recurrences with constant coefficients, signature (0, -3).

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]

concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

a(n) = -3*a(n-2) for n > 2.

E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024

cp's OEIS Frontend

A120580 Hankel transform of Sum_{k=0..n} C(2k,k).

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