A247170 Expansion of (-3/2+(x^3+3*x)/(sqrt(x^4-4*x^3-2*x^2+1)*2*x)).

0, 2, 3, 2, 10, 11, 21, 50, 66, 152, 275, 467, 988, 1717, 3283, 6386, 11560, 22556, 42465, 79832, 154122, 290039, 554323, 1060259, 2012310, 3859286, 7365423, 14072333, 26980788, 51580271, 98873291, 189567090, 363277676, 697348910

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

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

From R. J. Mathar, Jan 25 2020: (Start)

D-finite with recurrence: +3*n*a(n) +3*(n-1)*a(n-1) +(-5*n+2)*a(n-2) +(-17*n+25)*a(n-3) +(-11*n+34)*a(n-4) +(-3*n+25)*a(n-5) +(-3*n+20)*a(n-6) +(n-7)*a(n-7) = 0.

Conjectured: +n*(2*n-7)*a(n) +(n-1)*(2*n-9)*a(n-1) +2*(-2*n^2+9*n-6)*a(n-2) +2*(-6*n^2+33*n-38)*a(n-3) +3*(-2*n^2+15*n-26)*a(n-4) +(2*n-5)*(n-5)*a(n-5) = 0.

(End)

Mathematica

Table[n*Sum[(Binomial[k, n-2k]Binomial[n-k-1, k-1])/k, {k, n}], {n, 40}] (* Harvey P. Dale, Oct 04 2017 *)

Prog

(Maxima)
a(n):=n*sum((binomial(k, n-2*k)*binomial(n-k-1, k-1))/k, k, 1, n);
(Magma) R<x>:=PowerSeriesRing(Rationals(), 36); [0] cat Coefficients(R!( (-3/2+(x^3+3*x)/(Sqrt(x^4-4*x^3-2*x^2+1)*2*x)))); // Marius A. Burtea, Feb 11 2020
(Magma) [n*&+[Binomial(k, n-2*k)*Binomial(n-k-1, k-1)/k:k in [1..n]]:n in [1..35]]; // Marius A. Burtea, Feb 11 2020

Crossrefs

Cf. A025250.

Sequence in context: A075076 A333129 A078828 * A241055 A224418 A220947

Adjacent sequences: A247167 A247168 A247169 * A247171 A247172 A247173

Keyword

nonn

Author

Vladimir Kruchinin, Nov 21 2014

Status

approved