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A129442
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Expansion of c(x)*c(x*c(x)) where c(x) is the g.f. of A000108.
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8
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1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800
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OFFSET
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0,2
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COMMENTS
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The sequence b(n) = [0,1,2,6,21,80,322,1348,...] for n >= 0 is the Catalan transform of Catalan numbers C(n-1), with C(-1)=0; Sum_{k=0..n} A106566(n,k) * A000108(k-1) = b(n).
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LINKS
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FORMULA
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a(n-1) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1, n-1)*binomial(2*k-2, k-1).
G.f.: (1-sqrt(2*sqrt(1-4*x)-1))/(2*x). - Emeric Deutsch, Jun 20 2007 Corrected by Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007
Recurrence: 3*n*(n+1)*a(n) = 14*n*(2*n-1)*a(n-1) - 4*(4*n-5)*(4*n-3)*a(n-2).
a(n) ~ 2^(4*n+3/2)/(3^(n+1/2)*sqrt(Pi)*n^(3/2)). (End)
0 = +a(n)*(+a(n+1)*(+262144*a(n+2) -275968*a(n+3) +52608*a(n+4)) +a(n+2)*(-50176*a(n+2) +107680*a(n+3) -27930*a(n+4)) +a(n+3)*(-6006*a(n+3) +2574*a(n+4))) +a(n+1)*(+a(n+1)*(-17920*a(n+2) +21952*a(n+3) -4494*a(n+4)) +a(n+2)*(+5152*a(n+2) -15820*a(n+3) +4611*a(n+4)) +a(n+3)*(+1470*a(n+3) -630*a(n+4))) +a(n+2)*(+a(n+2)*(+42*a(n+2) +129*a(n+3) -63*a(n+4)) +a(n+3)*(-63*a(n+3) +27*a(n+4))) for n>=0. - Michael Somos, May 28 2023
G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2) ).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+k,k) * binomial(3*n-k+1,n-2*k). (End)
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EXAMPLE
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G.f. = 1 + 2*x + 6*x^2 + 21*x^3 + 80*x^4 + 322*x^5 + 1349*x^6 + ... - Michael Somos, May 28 2023
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MAPLE
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c := proc (x) options operator, arrow; (1/2)*(1-sqrt(1-4*x))/x end proc; G := simplify(c(x)*c(x*c(x))); Gser := series(G, x = 0, 28); seq(coeff(Gser, x, n), n = 0 .. 24) # Emeric Deutsch, Jun 20 2007
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MATHEMATICA
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a[n_]:= Sum[ Binomial[2n -k-1, n-1]*Binomial[2k-2, k-1], {k, n}]/n;
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PROG
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(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt(2*Sqrt(1-4*x)-1))/(2*x) )); // G. C. Greubel, Feb 06 2024
(SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-sqrt(2*sqrt(1-4*x)-1))/(2*x) ).list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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