A271180 Expansion of (4*x^3-7*x^2+4*x-1)/(x^6-4*x^5+4*x^4+x^3-7*x^2+5*x-1).

1, 1, 5, 15, 45, 125, 342, 921, 2461, 6535, 17282, 45567, 119898, 315020, 826830, 2168583, 5684731, 14896459, 39024899, 102216045, 267693813, 700997144, 1835543565, 4806092673, 12583591525, 32946281848, 86258240735, 225834015840

Offset: 0

Author

Vladimir Kruchinin, Apr 01 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-7,1,4,-4,1).

Crossrefs

Cf. A000045, 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) Fibonacci[k + 1], {i, 0, n - k}], {k, 0, n}], {n, 0, 27}] (* or *)
CoefficientList[Series[(4 x^3 - 7 x^2 + 4 x - 1)/(x^6 - 4 x^5 + 4 x^4 + x^3 - 7 x^2 + 5 x - 1), {x, 0, 27}], x] (* Michael De Vlieger, Apr 01 2016 *)

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)*fib(k+1),k,0,n);

PARI

x='x+O('x^99); Vec((4*x^3-7*x^2+4*x-1)/(x^6-4*x^5+4*x^4+x^3-7*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)*F(k+1)), where F = A000045 (Fibonacci numbers).

a(n) = 5*a(n-1) - 7*a(n-2) + a(n-3) + 4*a(n-4) - 4*a(n-5) + a(n-6) for n>3, a(0)=1, a(1)=1, a(2)=5, a(3)=15.