A181442 Expansion of (1 + x - 8*x^2 + x^3 + x^4) / ( (1 - x)*(1 - 10*x^2 + x^4) ).

1, 2, 4, 15, 35, 144, 342, 1421, 3381, 14062, 33464, 139195, 331255, 1377884, 3279082, 13639641, 32459561, 135018522, 321316524, 1336545575, 3180705675, 13230437224, 31485740222, 130967826661, 311676696541, 1296447829382, 3085281225184, 12833510467155

Offset: 0

Author

Paul Weisenhorn, Jan 29 2011

Keywords

• nonn

Comments

Previous name was: Solutions a(n) to (r(n)-2)*(r(n)-3) = 6*a(n)*(a(n)-1), where r(n) = A180483(n).

A combinatorial interpretation is provided in A180483.

Examples

			For n=3: a(3)=15; b(3)=38; binomial(38,4) = 73815 = binomial(38,2)*binomial(15,2).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1).

Crossrefs

Cf. A004189, A180483.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x-8*x^2+x^3+x^4)/((1-x)*(1-10*x^2+x^4)) )); // G. C. Greubel, Apr 26 2022

Maple

n:=0: for s from 1 to 100 do r:=(sqrt(24*s^2-24*s+1)+5)/2: if (floor(r)=r) then a[n]:=s: b[n]:=r: n:=n+1: end if: end do:

Mathematica

LinearRecurrence[{1,10,-10,-1,1},{1,2,4,15,35},30] (* Harvey P. Dale, Dec 22 2012 *)

SageMath

def b(n): return ((1+(-1)^n)/2)*chebyshev_U(n//2, 5)
def A181442(n): return (b(n) + 3*b(n-1) - 3*b(n-2) - b(n-3) + 1)/2
[A181442(n) for n in (0..50)] # G. C. Greubel, Apr 26 2022

Formula

G.f.: (1+x-8*x^2+x^3+x^4) / ( (1-x)*(1-10*x^2+x^4) ). - R. J. Mathar, Feb 05 2011

Explicit formulas: r=sqrt(6), s=5+2*r, t=5-2*r.

a(2*n) = (12 + (6 + r)*s^n + (6 - r)*t^n)/24.

a(2*n+1) = (12 + (18 + 7*r)*s^n + (18 - 7*r)*t^n)/24.

a(n) = 11*a(n-2) - 11*a(n-4) + a(n-6).

a(n) = +a(n-1) +10*a(n-2) -10*a(n-3) -a(n-4) +a(n-5).

a(n) = (b(n) + 3*b(n-1) - 3*b(n-2) - b(n-3) + 1)/2, where b(n) = ((1+(-1)^n)/2)* ChebyshevU(n/2, 5). - G. C. Greubel, Apr 26 2022

Extensions

New name using g.f. by R. J. Mathar from Joerg Arndt, Apr 27 2022