A239230 Expansion of -x*log'(-sqrt(12*x+2*sqrt(1-4*x)+2)/4+sqrt(1-4*x)/4+5/4).

0, 1, 1, 4, 9, 36, 112, 428, 1505, 5692, 21026, 79806, 301488, 1151866, 4403778, 16929474, 65204353, 251947668, 975366094, 3784197606, 14705937794, 57242631464, 223121176224, 870805992278, 3402485053664, 13308485156086, 52104519751272, 204176144516818

Offset: 0

Author

Vladimir Kruchinin, May 22 2014

Keywords

• nonn

Crossrefs

Cf. A055113.

Programs

Maple

a:= n-> add((-1)^(k+n)*binomial(2*n-k-1, n-1)*hypergeom([k-n, (n+1)/2, n/2], [n, n+1], 4), k=1..n);
seq(round(evalf(a(n), 32)), n=0..24); # Peter Luschny, May 22 2014

Maxima

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

Formula

G.f.: A(x) = x*F'(x)/(1-F(x)), where F(x) is g.f. of A055113.

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

a(n) = Sum_{k=1..n} (-1)^(k+n) * C(2*n-k-1,n-1) * hypergeom([k-n, n/2+1/2, n/2], [n, n+1], 4). - Peter Luschny, May 22 2014

Conjecture D-finite with recurrence +24*(365025561*n-1672569283)*(n-1)*(n-2)*(2*n-1)*a(n) -4*(n-2)*(27129169947*n^3-209577621466*n^2+463278020461*n-314084557758)*a(n-1) +2*(-28823853823*n^4+487259692534*n^3-3105214937957*n^2+8814274338098*n-9143920331436)*a(n-2) +4*(276083065830*n^4-4172118623320*n^3+24824880820695*n^2-69263721795041*n+75832154222148)*a(n-3) +(-587491214125*n^4+9941738070620*n^3-75070680472775*n^2+281912285021344*n-413197788157152)*a(n-4) +2*(-1186924847911*n^4+26108844767699*n^3-211936472383904*n^2+757584729548632*n-1009721693733312)*a(n-5) +4*(2*n-11)*(328530544924*n^3-5280431217363*n^2+28334632524947*n-50473913356356)*a(n-6) -72*(4143100547*n-18456753180)*(n-6)*(2*n-11)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Jul 27 2022

Conjecture: a(n) = Sum_{i=0..n-1} A059260(2*(n-1),n+i-1)*A009766(n+i-1,i)*(-1)^(n+i-1) = n*Sum_{i=0..n-1} binomial(n+2*i, i)/(n+2*i)*(-1)^(n+i-1)*[x^(n+i-1)] (1+(x+1)^(2*n-1))/(x+2) for n >= 0. - Mikhail Kurkov, Feb 18 2025