A290987 Expansion of (1 - 2*x + x^2 - x^4 + x^3 + x^5)/((1 - x)^2*(1 - 2*x + x^3 - x^4)).

1, 2, 4, 8, 16, 32, 63, 122, 233, 441, 830, 1557, 2915, 5451, 10186, 19026, 35529, 66337, 123849, 231211, 431631, 805768, 1504193, 2807986, 5241856, 9785309, 18266848, 34099850, 63656272, 118831031, 221829087, 414101780, 773028830, 1443059634, 2693846606

Offset: 0

Author

R. J. Mathar, Aug 16 2017

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 0..3684
• T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2, A(123;1,x).
• Index entries for linear recurrences with constant coefficients, signature (4,-5,1,3,-3,1).

Crossrefs

Cf. A059633, A290986, A290989.

Programs

Magma

I:=[1,2,4,8,16,32]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2) +Self(n-3)+3*Self(n-4)-3*Self(n-5)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Aug 17 2017

Maple

f:= gfun:-rectoproc({a(n)-3*a(n+1)+3*a(n+2)+a(n+3)-5*a(n+4)+4*a(n+5)-a(n+6), a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, a(4) = 16, a(5) = 32}, a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 16 2017

Mathematica

DeleteCases[#, 0] &@ CoefficientList[Series[(1-2x+x^2-x^4+x^3+x^5)/((1-x)^2*(1-2x +x^3-x^4)), {x,0,34}], x] (* Michael De Vlieger, Aug 16 2017 *)
LinearRecurrence[{4,-5,1,3,-3,1}, {1,2,4,8,16,32}, 40] (* Vincenzo Librandi, Aug 17 2017 *)

PARI

Vec((1-2*x+x^2-x^4+x^3+x^5)/((1-x)^2*(1-2*x+x^3-x^4)) + O(x^50)) \\ Michel Marcus, Aug 17 2017

SageMath

def A290987_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x+x^2-x^4+x^3+x^5)/((1-x)^2*(1-2*x+x^3-x^4)) ).list()
A290987_list(50) # G. C. Greubel, Apr 12 2023

Formula

a(n) = A059633(n+2) + A059633(n+3) - n.