A219753 Expansion of x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)).

0, 0, 0, 0, 1, 2, 6, 13, 31, 68, 151, 327, 706, 1509, 3211, 6796, 14331, 30115, 63106, 131905, 275111, 572676, 1190031, 2469039, 5115426, 10584557, 21874931, 45158908, 93131731, 191884523, 395000226, 812445097, 1669750831, 3429176564, 7037627911, 14433683991

Offset

0,6

Links

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

M. H. Albert, M. D. Atkinson and Robert Brignall, The enumeration of three pattern classes, arXiv:1206.3183 [math.CO], 2012.

Index entries for linear recurrences with constant coefficients, signature (4,-2,-7,4,4).

Formula

G.f.: x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)) (half of A219752).

a(n) = (2^(n-5)*(3*n+16)+4*(-1)^n)/9+((1-sqrt(5))^(n-3)-(1+sqrt(5))^(n-3))/(2^(n-3)*sqrt(5)) with n>3, a(0)=a(1)=a(2)=a(3)=0. [Bruno Berselli, Nov 29 2012]

a(n) = A219751(n)-A000045(n-3), n>=4. - R. J. Mathar, Aug 19 2022

Maple

A219753 := proc(n)
    if n < 4 then
        0 ;
    else
        128*(-1)^n+2^n*(16+3*n)-288*A000045(n-3) ;
        %/288 ;
    end if ;
end proc:
seq(A219753(n), n=0..20) ; # R. J. Mathar, Aug 19 2022

Mathematica

CoefficientList[Series[x^4 (1 - 2 x + x^4)/((1 + x) (1 - 2 x)^2 (1 - x - x^2)), {x, 0, 35}], x] (* Bruno Berselli, Nov 30 2012 *)
LinearRecurrence[{4, -2, -7, 4, 4}, {0, 0, 0, 0, 1, 2, 6, 13, 31}, 40] (* Harvey P. Dale, Oct 05 2021 *)

Prog

(Maxima) makelist(coeff(taylor(x^4*(1-2*x+x^4)/((1+x)*(1-2*x)^2*(1-x-x^2)), x, 0, n), x, n), n, 0, 35); /* Bruno Berselli, Nov 29 2012 */
(Magma) I:=[0, 0, 0, 0, 1, 2, 6, 13, 31]; [n le 9 select I[n] else 4*Self(n-1) - 2*Self(n-2) - 7*Self(n-3) + 4*Self(n-4) + 4*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 14 2012

Crossrefs

Cf. A219751-A219759, A219837.

Sequence in context: A094687 A369584 A336875 * A239305 A018013 A263899

Adjacent sequences: A219750 A219751 A219752 * A219754 A219755 A219756

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 28 2012

Status

approved