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A321398 a(n) = (-1)^(n+1)*n!* [x^n](log(x + 1)/2 + log(3*x + 1)/6). 2

%I

%S 0,1,2,10,84,984,14640,262800,5513760,132289920,3571464960,

%T 107140320000,3535590643200,127280784153600,4963944354969600,

%U 208485575730432000,9381849600195072000,450328759886573568000,22966766398527823872000,1240205379118128783360000

%N a(n) = (-1)^(n+1)*n!* [x^n](log(x + 1)/2 + log(3*x + 1)/6).

%H Robert Israel, <a href="/A321398/b321398.txt">Table of n, a(n) for n = 0..381</a>

%F E.g.f.: -log(1 - x)/2 - log(1 - 3*x)/6. - _Andrew Howroyd_, Nov 10 2018

%F 3*n*(n+1)*a(n)-4*(n+1)*a(n)+a(n+2)=0. - _Robert Israel_, Nov 10 2018

%p ser := series(ln(x+1)/2 + ln(1+3*x)/6, x, 21):

%p seq((-1)^(n+1)*n!*coeff(ser, x, n), n=0..19);

%t CoefficientList[Series[Log[x+1]/2 + Log[1+3*x]/6, {x, 0, 50}], x]* Table[(-1)^(n+1)*n!, {n, 0, 50}] (* _Stefano Spezia_, Nov 10 2018 *)

%o (PARI) seq(n)={Vec(serlaplace(-log(1 - x + O(x^n))/2 - log(1 - 3*x + O(x^n))/6), -n)} \\ _Andrew Howroyd_, Nov 10 2018

%o (MAGMA) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( -Log((1-x)^3*(1-3*x))/6 )); [0] cat [Factorial(n-0)*b[n]: n in [1..(m-1)]]; // _G. C. Greubel_, Nov 11 2018

%Y Cf. A133942 (n=1), A000165 (n=2), this sequence (n=3), A320962 (limit).

%K nonn

%O 0,3

%A _Peter Luschny_, Nov 10 2018

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Last modified June 24 05:21 EDT 2019. Contains 324318 sequences. (Running on oeis4.)