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G.f.: 2 - x*2/(1 - (1-8*x)^(1/4)).
1

%I #22 Nov 13 2024 05:37:32

%S 1,3,5,20,101,572,3470,22040,144669,973356,6676186,46503080,328034226,

%T 2338460056,16819478972,121903180848,889396747869,6526715628492,

%U 48141140144546,356708675726088,2653863473928870,19816831149068360,148466651633265540,1115659552758534480

%N G.f.: 2 - x*2/(1 - (1-8*x)^(1/4)).

%H G. C. Greubel, <a href="/A256093/b256093.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (Sum_{k=0..(n+1)} binomial(2*k-2,k)*2^(n-k+1)*binomial(2*n-k,n-k+1))/n, a(0)=1.

%F a(n) ~ 2^(3*n-4) / (Gamma(3/4) * n^(5/4)) * (1 + 2*Gamma(3/4) / (n^(1/4) * sqrt(Pi)) + 3*Gamma(3/4)^2 / (Pi*sqrt(2*n))). - _Vaclav Kotesovec_, Mar 15 2015

%F (512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3) = 0 for n >= 1. - _Robert Israel_, Jan 20 2020

%p f:= gfun:-rectoproc({(512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3), a(0) = 1, a(1) = 3, a(2) = 5, a(3) = 20},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Jan 20 2020

%t CoefficientList[Series[2-x*2/(1-(1-8*x)^(1/4)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 15 2015 *)

%o (Maxima)

%o a(n):=if n=0 then 1 else sum(binomial(2*k-2,k)*2^(n-k+1)*binomial(2*n-k,n-k+1),k,0,n+1)/n;

%o (PARI) my(x='x+O('x^50)); Vec(2-x*2/(1-(1-8*x)^(1/4))) \\ _G. C. Greubel_, Jun 03 2017

%K nonn,changed

%O 0,2

%A _Vladimir Kruchinin_, Mar 14 2015