A205492 Expansion of (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)).

1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976, 87369, 203915, 471546, 1082849, 2473535, 5627684, 12765052, 28887838, 65260270, 147233926, 331842395, 747355066, 1682185342, 3784718431, 8512408455, 19141037360, 43032743620

Offset: 0

Author

L. Edson Jeffery, Jan 28 2012

Keywords

• nonn

Comments

See array A205497 regarding association of this sequence with generating functions for the rows of the array form of A050446.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• L. E. Jeffery, Unit-primitive matrices
• Index entries for linear recurrences with constant coefficients, signature (7,-17,12,15,-26,3,13,-5,-2,1).

Crossrefs

Cf. A050446, A050447, A205497.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)) )); // G. C. Greubel, Jan 04 2020

Maple

seq(coeff(series((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 04 2020

Mathematica

LinearRecurrence[{7,-17,12,15,-26,3,13,-5,-2,1},{1,7,31,109,334,937,2475,6267, 15393,36976},30] (* Harvey P. Dale, Mar 26 2013 *)
CoefficientList[Series[(1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), {x,0,30}], x] (* G. C. Greubel, Jan 04 2020 *)

PARI

my(x='x+O('x^30)); Vec((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3))) \\ G. C. Greubel, Jan 04 2020

Sage

def A205492_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)) ).list()
A205492_list(30) # G. C. Greubel, Jan 04 2020

Formula

a(n) = 7*a(n-1) - 17*a(n-2) + 12*a(n-3) + 15*a(n-4) - 26*a(n-5) + 3*a(n-6) + 13*a(n-7) - 5*a(n-8) - 2*a(n-9) + a(n-10), n>9, {a(m)} = {1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976}, m=0,...,9.

CONJECTURE 1. a(n) = M_{n,2} = M_{2,n}, where M = A205497.

CONJECTURE 2. lim_{n->oo} a(n+1)/a(n) = (2*cos(Pi/7))^2-1 = A116425-1 = spectral radius of the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,2} = [0,0,1; 0,1,1; 1,1,1].