A081583 Third row of Pascal-(1,2,1) array A081577.

1, 10, 46, 136, 307, 586, 1000, 1576, 2341, 3322, 4546, 6040, 7831, 9946, 12412, 15256, 18505, 22186, 26326, 30952, 36091, 41770, 48016, 54856, 62317, 70426, 79210, 88696, 98911, 109882, 121636, 134200, 147601, 161866, 177022, 193096

Offset: 0

Author

Paul Barry, Mar 23 2003

Keywords

• easy
• nonn

Comments

Equals binomial transform of [1, 9, 27, 27, 0, 0, 0, ...] where (1, 9, 27, 27) = row 3 of triangle A013610. - Gary W. Adamson, Jul 19 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A013610, A038764, A081584, A081577.

Programs

Magma

[(2+9*n+9*n^3)/2: n in [0..40]]; // Vincenzo Librandi, Aug 09 2013

Maple

seq((2+9*n+9*n^3)/2, n=0..40); # G. C. Greubel, May 25 2021

Mathematica

CoefficientList[Series[(1+2x)^3/(1-x)^4, {x,0,50}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,10,46,136},60] (* Harvey P. Dale, Oct 01 2021 *)

Sage

a = lambda n: hypergeometric([-n, -3], [1], 3)
[simplify(a(n)) for n in range(36)] # Peter Luschny, Nov 19 2014

Formula

a(n) = (2 + 9*n + 9*n^3)/2.

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

a(n) = hypergeommetric2F1([-n, -3], [1], 3). - Peter Luschny, Nov 19 2014

E.g.f.: (1/2)*(2 + 18*x + 27*x^2 + 9*x^3)*exp(x). - G. C. Greubel, May 25 2021