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Expansion of e.g.f. -log(1 - x - x^2/2).
0

%I #14 Mar 06 2022 09:45:46

%S 1,2,5,21,114,780,6390,61110,667800,8210160,112152600,1685237400,

%T 27624920400,490572482400,9381882510000,192238348302000,

%U 4201639474032000,97572286427616000,2399151995223984000,62268748888378032000,1701213856860117600000

%N Expansion of e.g.f. -log(1 - x - x^2/2).

%F a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.

%F D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - _R. J. Mathar_, Aug 20 2021

%p b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] end:

%p a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(

%p binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)

%p end:

%p seq(a(n), n=1..25); # _Alois P. Heinz_, Oct 11 2019

%t nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%t FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ _Michel Marcus_, Oct 11 2019

%Y Cf. A009014, A039647, A080040, A080599 (exponential transform).

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Oct 11 2019