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A323230 a(n) = binomial(2*(n - 1), n - 1) + 1.

%I A323230 #12 Oct 23 2023 11:25:07
%S A323230 1,2,3,7,21,71,253,925,3433,12871,48621,184757,705433,2704157,
%T A323230 10400601,40116601,155117521,601080391,2333606221,9075135301,
%U A323230 35345263801,137846528821,538257874441,2104098963721,8233430727601,32247603683101,126410606437753,495918532948105
%N A323230 a(n) = binomial(2*(n - 1), n - 1) + 1.
%H A323230 G. C. Greubel, <a href="/A323230/b323230.txt">Table of n, a(n) for n = 0..500</a>
%F A323230 Let G(x) = -1/(x - 1) - I*x/sqrt(4*x - 1) with Im(x) < 0, then a(n) = [x^n] G(x).
%F A323230 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).
%F A323230 a(n) = ((4*n - 6)*a(n - 1) - 3*n + 5)/(n - 1) for n >= 2. - _Peter Luschny_, Aug 02 2019
%F A323230 From _G. C. Greubel_, Dec 09 2021: (Start)
%F A323230 a(n) = 1 + n*A000108(n-1).
%F A323230 E.g.f.: exp(x) + x*exp(2*x)*(BesselI[0, 2*x] - BesselI[1, 2*x]). (End)
%p A323230 aList := proc(len) local gf, ser; assume(Im(x)<0);
%p A323230 gf := -1/(x-1) - I*x/sqrt(4*x-1); ser := series(gf, x, len+2):
%p A323230 seq(coeff(ser, x, n), n=0..len) end: aList(27);
%p A323230 # Alternative:
%p A323230 a := proc(n) option remember;
%p A323230 if n < 2 then [1, 2][n+1] else ((4*n - 6)*a(n - 1) - 3*n + 5)/(n - 1) fi end:
%p A323230 seq(a(n), n=0..27); # _Peter Luschny_, Aug 02 2019
%t A323230 Table[Binomial[2(n - 1), n - 1] + 1, {n, 0, 27}]
%o A323230 (Magma) [1] cat [1 + n*Catalan(n-1): n in [1..30]]; // _G. C. Greubel_, Dec 09 2021
%o A323230 (Sage) [1 + binomial(2*n-2, n-1) for n in (0..30)] # _G. C. Greubel_, Dec 09 2021
%o A323230 (PARI) a(n)=binomial(2*n-2, n-1)+1 \\ _Charles R Greathouse IV_, Oct 23 2023
%Y A323230 Compare to A244174 which is "missing" the second term 2.
%Y A323230 Cf. A000108.
%K A323230 nonn,easy
%O A323230 0,2
%A A323230 _Peter Luschny_, Feb 12 2019