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A179533
Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.
0
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
0,2
COMMENTS
Hankel transform is A168495(n+1).
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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Barry, Jan 08 2011
STATUS
approved