A080952 a(n) = 2^(n-4)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)/15.

3, 21, 112, 504, 2016, 7392, 25344, 82368, 256256, 768768, 2236416, 6336512, 17547264, 47628288, 127008768, 333398016, 862912512, 2205220864, 5571084288, 13927710720, 34487664640, 84651540480, 206108098560, 498094571520

Offset

0,1

Comments

Old definition was "Sequence associated with recurrence a(n) = 2*a(n-1) + k*(k+2)*a(n-2)". See the first comment in A080928.

The sixth column of A080928 (after 0) is 2*a(n).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).

Formula

G.f.: (1-x)*(4*x^2-2*x+1)*(4*x^2-6*x+3)/(1-2x)^6.

a(n) = 12*a(n-1) - 60*a(n-2) + 160*a(n-3) - 240*a(n-4) + 192*a(n-5) - 64*a(n-6), n>=6. - Harvey P. Dale, Jun 11 2011

Let b(n) = A000292(n+1)+n+1+A000389(n+5) = (n+1)*(n^4+14*n^3+91*n^2+254*n+360)/120 = 3, 12, 34, 80, 166, 314,.. Then a(n) = 2^n*b(n) - 2^(n-1)*b(n-1). - R. J. Mathar, Jun 11 2011

From Amiram Eldar, Jan 07 2022: (Start)

Sum_{n>=0} 1/a(n) = 40*log(2) - 82/3.

Sum_{n>=0} (-1)^n/a(n) = 1314 - 3240*log(3/2). (End)

Mathematica

LinearRecurrence[{12, -60, 160, -240, 192, -64}, {3, 21, 112, 504, 2016, 7392}, 30] (* or *) CoefficientList[Series[(1-x) (3 - 12 x + 28 x^2 - 32 x^3 + 16 x^4)/ (1 - 2 x)^6, {x, 0, 30}], x] (* Harvey P. Dale, Jun 11 2011 *)

Prog

(Magma) I:=[3, 21, 112, 504, 2016, 7392]; [n le 6 select I[n] else 12*Self(n-1)-60*Self(n-2)+160*Self(n-3)-240*Self(n-4)+192*Self(n-5)-64*Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 06 2013
(PARI) my(x='x+O('x^50)); Vec((1-x)*(4*x^2-2*x+1)*(4*x^2-6*x+3)/(1-2*x)^6) \\ G. C. Greubel, Nov 24 2017

Crossrefs

Cf. A000292, A000389, A080928.

Sequence in context: A233582 A043012 A122120 * A183404 A309670 A121140

Adjacent sequences: A080949 A080950 A080951 * A080953 A080954 A080955

Keyword

nonn,easy

Author

Paul Barry, Feb 26 2003

Extensions

Replaced the previous definition with the closed form from Bruno Berselli, Aug 06 2013

Status

approved