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

0,4

Links

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

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

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

Prog

(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);

Crossrefs

Cf. A055113.

Sequence in context: A149138 A149139 A149140 * A149141 A149142 A149143

Adjacent sequences: A239227 A239228 A239229 * A239231 A239232 A239233

Keyword

nonn

Author

Vladimir Kruchinin, May 22 2014

Status

approved