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A321849 Expansion of e.g.f.: exp(x/(1-6*x)).

%I A321849 #33 Sep 08 2022 08:46:23
%S A321849 1,1,13,253,6553,211801,8201701,369979093,19047250993,1101705494833,
%T A321849 70715424362941,4987040544656941,383243311962126793,
%U A321849 31871863566298601353,2851588139929576342933,273093945561709965890821,27871997808801906673665121
%N A321849 Expansion of e.g.f.: exp(x/(1-6*x)).
%C A321849 For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.
%H A321849 Ludovic Schwob, <a href="/A321849/b321849.txt">Table of n, a(n) for n = 0..199</a>
%F A321849 a(n) = Sum_{k=0..n} 6^(n-k)*(n!/k!)*binomial(n-1, k-1).
%F A321849 Recurrence: a(n) = (12*n-11)*a(n-1) - 36*(n-2)*(n-1)*a(n-2).
%F A321849 a(n) ~ n! * exp(sqrt(2*n/3) - 1/12) * 6^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Nov 21 2018
%p A321849 seq(coeff(series(factorial(n)*exp(x/(1-6*x)),x,n+1), x, n), n = 0 .. 17); # _Muniru A Asiru_, Nov 24 2018
%t A321849 a[n_] := Sum[6^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (12n - 11)*a[n - 1] - 36(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* _Amiram Eldar_, Nov 19 2018 *)
%o A321849 (PARI) my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-6*x)))) \\ _Michel Marcus_, Nov 25 2018
%o A321849 (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x/(1-6*x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Dec 08 2018
%o A321849 (Sage)
%o A321849 f= exp(x/(1-6*x))
%o A321849 g=f.taylor(x,0,13)
%o A321849 L=g.coefficients()
%o A321849 coeffs={c[1]:c[0]*factorial(c[1]) for c in L}
%o A321849 coeffs  # _G. C. Greubel_, Dec 08 2018
%o A321849 (Magma) [1] cat [&+[6^(n-k)* Factorial(n)/Factorial(k)*Binomial(n-1, k-1):  k in [0..n]]: n in [1.. 18]]; // _Vincenzo Librandi_, Dec 08 2018
%Y A321849 Cf. A000262, A025168, A321837, A321847, A321848, A321850.
%K A321849 nonn
%O A321849 0,3
%A A321849 _Ludovic Schwob_, Nov 19 2018