A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.

1, 2, 7, 23, 85, 332, 1369, 5870, 25945, 117374, 540805, 2528675, 11966923, 57206972, 275824159, 1339721519, 6549093013, 32195473406, 159065828029, 789395034701, 3933239089903, 19668745466636, 98679891233803, 496570499905832, 2505670304785615, 12675395921692394, 64270076976110203, 326580624341708693, 1662796531746045157, 8481930651824392268, 43341418581113085697

Offset: 0

Author

Paul Barry, Jan 08 2011

Keywords

• nonn

Comments

Hankel transform is A168495(n+1).

Crossrefs

Cf. A000108, A073370, A168495.

Programs

Maple

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1/(1-x-2*x^2))*(1 - sqrt(1 - 4*(x/(1-x-2*x^2)))) / (2*(x/(1-x-2*x^2))):
init:= [1, 2, 7, 23, 85, 332, 1369];
iseq:= seq(u(i-1)=init[i],i=1..nops(init)): req:= FindRE(ogf,x,u(n));
rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..30); # Georg Fischer, Nov 04 2022

Formula

G.f.: 1/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-... (continued fraction);

a(n) = Sum_{k=0..n} A000108(k)*Sum_{j=0..n-k} C(k+j,k)*C(j,n-k-j)*2^(n-k-j).

a(n) = Sum_{k=0..n} A073370(n,k)*A000108(k).

D-finite with recurrence: (n+1)*a(n) +2*(1-3n)*a(n-1) +(n-1)*a(n-2) +4*(3n-5)*a(n-3) +4*(n-3)*a(n-4)= 0. - R. J. Mathar, Nov 17 2011