|
%I #21 Feb 14 2024 04:04:00
%S 0,0,0,0,2,20,120,567,2320,8596,29578,96058,297776,888460,2567534,
%T 7221974,19849284,53473464,141559090,369018290,948897336,2410350276,
%U 6055659350,15063123950,37130340492,90769197360,220209351130,530489281162,1269657766720
%N Expansion of x^4*(2-10*x+18*x^2-7*x^3-21*x^4+25*x^5-x^6)/((1-x)^3*(1-2*x)^6).
%H Bruno Berselli, <a href="/A219837/b219837.txt">Table of n, a(n) for n = 0..1000</a>
%H M. H. Albert, M. D. Atkinson and Robert Brignall, <a href="http://arxiv.org/abs/1206.3183">The enumeration of three pattern classes</a>, arXiv:1206.3183 [math.CO] (2012), p. 28.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (15,-99,377,-912,1452,-1520,1008,-384,64).
%F G.f.: x^4*(2-12*x+28*x^2-25*x^3-14*x^4+46*x^5-26*x^6+x^7)/((1-x)^4*(1-2*x)^6) = x^4*(2-10*x+18*x^2-7*x^3-21*x^4+25*x^5-x^6)/((1-x)^3*(1-2*x)^6) (the first formula is in the paper indicated in Link section).
%F a(n) = 2^(n-10)*(n^5-16*n^4+547*n^3-7976*n^2+55660*n-158832)/3+(3*n+17)*n+52 with n>1, a(0)=a(1)=0.
%t CoefficientList[Series[x^4 (2 - 10 x + 18 x^2 - 7 x^3 - 21 x^4 + 25 x^5 - x^6)/((1 - x)^3 (1 - 2 x)^6), {x, 0, 28}], x]
%t LinearRecurrence[{15,-99,377,-912,1452,-1520,1008,-384,64},{0,0,0,0,2,20,120,567,2320,8596,29578},30] (* _Harvey P. Dale_, Dec 03 2020 *)
%o (Maxima) makelist(coeff(taylor(x^4*(2-10*x+18*x^2-7*x^3-21*x^4+25*x^5-x^6)/((1-x)^3*(1-2*x)^6), x, 0, n), x, n), n, 0, 28);
%o (Magma) I:=[0, 0, 0, 0, 2, 20, 120, 567, 2320, 8596, 29578, 96058, 297776]; [n le 11 select I[n] else 15*Self(n-1) - 99*Self(n-2) + 377*Self(n-3) - 912*Self(n-4) + 1452*Self(n-5) - 1520*Self(n-6) + 1008*Self(n-7) - 384*Self(n-8) + 64*Self(n-9): n in [1..30]]; // _Vincenzo Librandi_, Dec 14 2012
%Y Cf. A219751-A219759.
%K nonn,easy
%O 0,5
%A _Bruno Berselli_, Nov 29 2012
|
