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
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).
Programs
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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 *)
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
Comments