A270595 a(n) = (n+1)*Sum_{i=0..n/2}((binomial(n-i,i))*binomial(2*(n-2*i),(n-2*i))/(n-2*i+1)^2).

1, 1, 5, 9, 29, 81, 265, 881, 3065, 10897, 39533, 145561, 542645, 2043745, 7764497, 29719521, 114495985, 443623521, 1727552085, 6757763753, 26541775821, 104626427441, 413800025625, 1641535159889, 6529945089769, 26041812710001

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

G.f.: ((x^2+1)*(1-sqrt(1-(4*x)/(1-x^2))))/(2*x*(1-x^2)).

a(n) ~ 5^(1/4) * (5-2*sqrt(5)) * (2+sqrt(5))^(n+2) / (2^(7/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2016. Equivalently, a(n) ~ 5^(3/4) * phi^(3*n + 3) / (2^(7/2) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

From Benedict W. J. Irwin, May 29 2016: (Start)

Let y1(0)=0, y1(1)=1, y1(2)=-1, y1(3)=-3, y1(4)=-9,

Let -n*y1(n)-(3n+6)*y1(n+1)+(6n+10)*y1(n+2)+(2n+6)*y1(n+3)-(5n+14)*y1(n+4)+(n+4)*y1(n+5) = 0,

Let y2(-3)=0, y2(-2)=0, y2(-1)=0, y2(0)=0, y2(1)=1,

Let -n*y2(n)+(5n+6)*y2(n+1)-(2n+2)*y2(n+2)-(6n+14)*y2(n+3)+(3n+6)*y2(n+4)+(n+4)*y2(n+5) = 0,

a(n) = (2-y1(n)-y1(n+2)+(-1)^n*(y2(n)+y2(n+2)-2))/4.

(End)

Mathematica

Q[k_] := DifferenceRoot[Function[{y, n}, {-n y[n] + (-6 - 3 n) y[1 + n] + (10 + 6 n) y[2 + n] + (6 + 2 n) y[3 + n] + (-14 - 5 n) y[4 + n] + (4 + n) y[5 + n] == 0, y[0] == 0, y[1] == 1, y[2] == -1, y[3] == -3, y[4] == -9}]][k];
P[k_] := DifferenceRoot[Function[{y, n}, {-n y[n] + (6 + 5 n) y[1 + n] + (-2 - 2 n) y[2 + n] + (-14 - 6 n) y[3 + n] + (6 + 3 n) y[4 + n] + (4 + n) y[5 + n] == 0, y[-3] == 0, y[-2] == 0, y[-1] == 0, y[0] == 0, y[1] == 1}]][k];
Table[1/4 (2 - 2 (-1)^n - Q[n] - Q[2 + n] + (-1)^n P[n] + (-1)^n P[2 + n]), {n, 0, 20}] (* Benedict W. J. Irwin, May 29 2016 *)
CoefficientList[Series[((x^2 + 1)*(1 - Sqrt[1 - (4*x)/(1 - x^2)])) / (2*x*(1 - x^2)), {x, 0, 50}], x] (* G. C. Greubel, May 29 2016 *)

Prog

(Maxima)
a(n):=(n+1)*sum((binomial(n-i, i))*binomial(2*(n-2*i), (n-2*i))/(n-2*i+1)^2, i, 0, (n)/2);
(PARI) x='x+O('x^100); Vec(((x^2+1)*(1-sqrt(1-(4*x)/(1-x^2))))/(2*x*(1-x^2))) \\ Altug Alkan, Mar 20 2016

Crossrefs

Cf. A000108.

Sequence in context: A193487 A006131 A352008 * A303988 A242329 A343420

Adjacent sequences: A270592 A270593 A270594 * A270596 A270597 A270598

Keyword

nonn

Author

Vladimir Kruchinin, Mar 20 2016

Status

approved