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A323230
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a(n) = binomial(2*(n - 1), n - 1) + 1.
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6
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1, 2, 3, 7, 21, 71, 253, 925, 3433, 12871, 48621, 184757, 705433, 2704157, 10400601, 40116601, 155117521, 601080391, 2333606221, 9075135301, 35345263801, 137846528821, 538257874441, 2104098963721, 8233430727601, 32247603683101, 126410606437753, 495918532948105
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OFFSET
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0,2
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LINKS
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FORMULA
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Let G(x) = -1/(x - 1) - I*x/sqrt(4*x - 1) with Im(x) < 0, then a(n) = [x^n] G(x).
The generating function G(x) satisfies the differential equation 6*x^2 - 4*x + 1 = (4*x^4 - 9*x^3 + 6*x^2 - x)*diff(G(x), x) - (2*x^3 - 5*x^2 + 4*x - 1)*G(x).
a(n) = ((4*n - 6)*a(n - 1) - 3*n + 5)/(n - 1) for n >= 2. - Peter Luschny, Aug 02 2019
E.g.f.: exp(x) + x*exp(2*x)*(BesselI[0, 2*x] - BesselI[1, 2*x]). (End)
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MAPLE
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aList := proc(len) local gf, ser; assume(Im(x)<0);
gf := -1/(x-1) - I*x/sqrt(4*x-1); ser := series(gf, x, len+2):
seq(coeff(ser, x, n), n=0..len) end: aList(27);
# Alternative:
a := proc(n) option remember;
if n < 2 then [1, 2][n+1] else ((4*n - 6)*a(n - 1) - 3*n + 5)/(n - 1) fi end:
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MATHEMATICA
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Table[Binomial[2(n - 1), n - 1] + 1, {n, 0, 27}]
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PROG
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(Magma) [1] cat [1 + n*Catalan(n-1): n in [1..30]]; // G. C. Greubel, Dec 09 2021
(Sage) [1 + binomial(2*n-2, n-1) for n in (0..30)] # G. C. Greubel, Dec 09 2021
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CROSSREFS
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Compare to A244174 which is "missing" the second term 2.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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