A247172 Expansion of -1 - (sqrt(x^4 + 4*x^3 - 2*x^2 - 4*x + 1) - x^2 - 2*x - 1)/(4*x).

0, 1, 1, 3, 9, 29, 97, 335, 1185, 4273, 15649, 58051, 217673, 823709, 3141697, 12065087, 46613121, 181049025, 706549185, 2769069091, 10894034633, 43008192637, 170327978401, 676511841743, 2694115428769, 10755172567089, 43032673671137, 172538997381155

Offset

0,4

Links

Table of n, a(n) for n=0..27.

Formula

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

D-finite with recurrence: (n+1)*a(n) +2*(-2*n+1)*a(n-1) +2*(-n+2)*a(n-2) +2*(2*n-7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

From Emanuele Munarini, Jul 11 2024: (Start)

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n+k-1,3*k)*2^k*Catalan(k).

a(n) ~ (1/2)*sqrt(sqrt(5)/(2*Pi))*(2+sqrt(5))^n/n^(3/2). (End)

a(n) = hypergeom([1/2, 1/2 - n/2, 1 - n/2, n], [1/3, 2/3, 2], 32/27) for n > 0. -Peter Luschny, Jul 11 2024

Mathematica

CoefficientList[Series[-1-(Sqrt[x^4+4*x^3-2*x^2-4*x+1]-x^2-2*x-1)/(4*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 23 2014 *)
a[0] := 0;
a[n_] := HypergeometricPFQ[{1/2, (1 - n)/2, 1 - n/2, n}, {1/3, 2/3, 2}, 32/27];
Table[a[n], {n, 0, 26}]  (* Peter Luschny, Jul 11 2024 *)

Prog

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

Crossrefs

Cf. A000108.

Sequence in context: A124431 A071740 A081696 * A339843 A148939 A346158

Adjacent sequences: A247169 A247170 A247171 * A247173 A247174 A247175

Keyword

nonn,changed

Author

Vladimir Kruchinin, Nov 22 2014

Status

approved