A271176 Expansion of -(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1).

1, 1, 4, 13, 36, 94, 239, 597, 1471, 3586, 8669, 20818, 49726, 118259, 280239, 662117, 1560516, 3670321, 8617584, 20203698, 47308391, 110659649, 258614439, 603929562, 1409413761, 3287385206, 7664034874, 17860302403

Offset: 0

Author

Vladimir Kruchinin, Apr 01 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-2).

Crossrefs

Cf. A034008.

Programs

Mathematica

Table[(n + 1) Sum[Sum[(Binomial[i + k, i] 2^i Binomial[2 k + 2, n - i - k] (-1)^(n - i - k))/(k + 1), {i, 0, n - k}], {k, 0, n}], {n, 0, 27}] (* or *)
CoefficientList[Series[-(4 x^3 - 7 x^2 + 4 x - 1)/(2 x^4 - 5 x^3 + 8 x^2 - 5 x + 1), {x, 0, 27}], x] (* Michael De Vlieger, Apr 01 2016 *)
LinearRecurrence[{5,-8,5,-2},{1,1,4,13},30] (* Harvey P. Dale, Jan 19 2021 *)

Maxima

a(n):=(n+1)*sum(sum(binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k),i,0,n-k)/(k+1),k,0,n);

PARI

x='x+O('x^99); Vec(-(4*x^3-7*x^2+4*x-1)/(2*x^4-5*x^3+8*x^2-5*x+1)) \\ Altug Alkan, Apr 01 2016

Formula

a(n) = (n+1)*Sum_{k=0..n} (Sum_{i=0..n-k} (binomial(i+k,i)*2^i*binomial(2*k+2,n-i-k)*(-1)^(n-i-k))/(k+1)).

a(n) = 5*a(n-1)-8*a(n-2)+5*a(n-3)-2*a(n-4) for n>3, a(0)=1, a(1)=1, a(2)=4, a(3)=13.