A174227 Expansion of -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).

1, 1, 12, 145, 1764, 21602, 266232, 3301349, 41178660, 516512462, 6513158376, 82542517386, 1051024082472, 13442267711940, 172638285341040, 2225824753934445, 28802104070304420, 373966734921011990

Offset: 0

Author

Paul Barry, Mar 12 2010

Keywords

• nonn
• easy

Comments

Hankel transform is A077417.

The g.f. A(x) satisfies the continued fraction relation A(x) = 1/(1-x/(1-10*x-x*A(x))).

Programs

Maple

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x): req:= FindRE(ogf,x,u(n));
init:= [1, 1, 12, 145]: iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
rmin:= subs(n=n-2, MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..17); # Georg Fischer, Nov 03 2022
# Alternative, using function FindSeq from A174403:
ogf := -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x):
a := FindSeq(ogf): seq(a(n), n=0..17); # Peter Luschny, Nov 04 2022

Formula

a(n) = sqrt(5/7) * 10^n * (6*hypergeom([1/2, n+1],[1],2/7)-7*hypergeom([1/2, n],[1],2/7)) / (n+1) for n > 0. - Mark van Hoeij, Jul 02 2010

D-finite with recurrence: (n+1)*a(n) +12*(1-2*n)*a(n-1) +140*(n-2)*a(n-2)=0. - R. J. Mathar, Sep 30 2012

Extensions

Definiton corrected by Peter Luschny, Nov 05 2022