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

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

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

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

From G. C. Greubel, Dec 09 2021: (Start)

a(n) = 1 + n*A000108(n-1).

E.g.f.: exp(x) + x*exp(2*x)*(BesselI[0, 2*x] - BesselI[1, 2*x]). (End)

Maple

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:
seq(a(n), n=0..27); # Peter Luschny, Aug 02 2019

Mathematica

Table[Binomial[2(n - 1), n - 1] + 1, {n, 0, 27}]

Prog

(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
(PARI) a(n)=binomial(2*n-2, n-1)+1 \\ Charles R Greathouse IV, Oct 23 2023

Crossrefs

Compare to A244174 which is "missing" the second term 2.

Cf. A000108.

Sequence in context: A047693 A212265 A107108 * A262305 A189360 A001532

Adjacent sequences: A323227 A323228 A323229 * A323231 A323232 A323233

Keyword

nonn,easy

Author

Peter Luschny, Feb 12 2019

Status

approved