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A071742
Expansion of (1+x^4*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
0
1, 1, 2, 5, 15, 44, 137, 443, 1472, 4994, 17225, 60216, 212874, 759696, 2733226, 9902857, 36100570, 132319230, 487333545, 1802620860, 6693765210, 24943905720, 93249826830, 349623734070, 1314370414344, 4953428965092
OFFSET
0,3
FORMULA
Recurrence: -(n+1)*(n-2)*a(n) +4*(2*n^2-7*n+1)*a(n-1) -4*(2*n-3)*(2*n-9)*a(n-2)=0. - R. J. Mathar, Jun 24 2018
MAPLE
with(FormalPowerSeries): with(LREtools): # requires Maple 2022
C:= (1-(1-4*x)^(1/2))/(2*x);
inits:= {a(0)=1, a(1)=1, a(2)=2, a(3)=5, a(4)=15, a(5)=44, a(6)=137, a(7)=443};
re:= FindRE((1+x^4*C)*C, x, a(n));
rm:= -(n+1)*(n-2)*a(n) +4*(2*n^2-7*n+1)*a(n-1) -4*(2*n-3)*(2*n-9)*a(n-2)=0;
minre:= MinimalRecurrence(re, a(n), inits);
minrm:= MinimalRecurrence(rm, a(n), inits);
# shows that Mathar's recurrence is equivalent
f:= REtoproc(re, a(n), inits);
seq(f(n), n=0..25); # Georg Fischer, Oct 22 2022
MATHEMATICA
Catl[x_] = (1 - (1 - 4*x)^(1/2))/(2*x);
Table[SeriesCoefficient[(1 + x^4*Catl[x])*Catl[x], {x, 0, n}], {n, 0, 25}] (* Hugo Pfoertner, Oct 22 2022 *)
CROSSREFS
Sequence in context: A084086 A307259 A292524 * A035355 A220873 A148351
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved