A137717 -id:A137717 - cp's OEIS frontend

A132310 a(n) = 3^nSum_{ k=0..n } binomial(2k,k)/3^k.

1, 5, 21, 83, 319, 1209, 4551, 17085, 64125, 240995, 907741, 3428655, 12990121, 49370963, 188229489, 719805987, 2760498351, 10615101273, 40920439119, 158106581157, 612166272291, 2374756691313, 9228369037659, 35918537840577

Offset: 0

Paul D. Hanna, Aug 18 2007

nonn

Simpler definition from N. J. A. Sloane, Jan 21 2009. Colin Mallows and I studied this sequence on Feb 21 1981 in connection with integration over a regular (solid) hexagon.

Hankel transform is A137717. - Paul Barry, Apr 26 2009

			a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5;
a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21;
a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) = 83.
2*a(6) = sum(A182411(7,i), i=0..6) = 3432+858+572+572+728+1092+1848 = 9102 = 2*4551. - _Bruno Berselli_, May 02 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..300

CoefficientList[Series[1/Sqrt[1-10*x+33*x^2-36*x^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n)=binomial(2*n,n)*sum(k=0,2*n, polcoeff((1+x+x^2)^n,k)/binomial(2*n,k) )

a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k) * k!*(2*n-k)! / (n!)^2 )

a(n) = C(2n,n) * sum_{k=0..2n} trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n, where [x^k] denotes "coefficient of x^k in ...".
G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).
a(n) = sum_{k=0..2n} trinomial(n,k) * k!*(2*n-k)! / (n!)^2.
2*a(n) = sum(A182411(n+1,i), i=0..n). - Bruno Berselli, May 02 2012
D-finite with recurrence: n*a(n) = (7*n-2)*a(n-1) - 6*(2*n-1)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 4^(n+1)/sqrt(Pi*n) . - Vaclav Kotesovec, Oct 20 2012

A137718 A scaled Hankel transform.

Original entry on oeis.org

1, -4, 2, 4, -8, 8, 8, -32, 16, 32, -64, 64, 64, -256, 128, 256, -512, 512, 512, -2048, 1024, 2048, -4096, 4096, 4096, -16384, 8192, 16384, -32768, 32768, 32768, -131072, 65536, 131072, -262144, 262144, 262144

Offset: 0

Paul Barry, Feb 08 2008

A137717=2^floor(n/2)*a(n) is the Hankel transform of A106191.

Cf. A137719.

a(n)=A137717(n)/2^floor(n/2).

Empirical g.f.: -(4*x^3-4*x^2+4*x-1) / (4*x^4+2*x^2+1). - Colin Barker, Jun 27 2013

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

A132310 a(n) = 3^nSum_{ k=0..n } binomial(2k,k)/3^k.

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A137718 A scaled Hankel transform.

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

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cp's OEIS Frontend

A132310 a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)/3^k.

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A137718 A scaled Hankel transform.

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Author

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Comments

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A132310 a(n) = 3^nSum_{ k=0..n } binomial(2k,k)/3^k.