A166336 Expansion of (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - 7*x + 17*x^2 - 17*x^3 + 7*x^4 - x^5).

1, 3, 11, 39, 131, 421, 1309, 3971, 11823, 34691, 100611, 289033, 823801, 2332419, 6566291, 18394911, 51310979, 142587181, 394905493, 1090444931, 3002921271, 8249479163, 22612505091, 61857842449, 168903452401, 460409998851

Offset

0,2

Comments

The diagonal sums of number triangle A166335 are 1, 0, 3, 0, 11, 0, ...

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-7,1).

Formula

G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/((1 - x)*(1 - 3*x + x^2)^2);

a(n) = 1 + 2*Sum{k=0..n} k*C(n + k, 2*k) = 1 + 2*Sum{k=0..n} (n-k)*C(2*n - k, k) = 1 + 2*A001870(n).

a(0) = 1, a(1) = 3, a(2) = 11, a(3) = 39, a(4) = 131, and a(n) = -17*a(n-1) + 17*a(n-2) - 7*a(n-3) + a(n-4) for n >= 4. - Harvey P. Dale, Jul 05 2014

Mathematica

CoefficientList[Series[(1-4x+7x^2-4x^3+x^4)/(1-7x+17x^2-17x^3+7x^4-x^5), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -17, 17, -7, 1}, {1, 3, 11, 39, 131}, 30] (* Harvey P. Dale, Jul 05 2014 *)

Crossrefs

Cf. A001870, A166335.

Sequence in context: A064086 A089579 A227638 * A002783 A289834 A007482

Adjacent sequences: A166333 A166334 A166335 * A166337 A166338 A166339

Keyword

easy,nonn

Author

Paul Barry, Oct 12 2009

Status

approved