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Expansion of e.g.f.: exp(x/(1-3*x)).
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%I #41 Sep 08 2022 08:46:23

%S 1,1,7,73,1009,17341,355951,8488117,230439553,7013527129,236419161751,

%T 8740611892321,351566026652017,15280473017519893,713558666964639679,

%U 35623071889296787981,1893073661362838712961,106682309871314293118257

%N Expansion of e.g.f.: exp(x/(1-3*x)).

%C For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.

%H Ludovic Schwob, <a href="/A321837/b321837.txt">Table of n, a(n) for n = 0..199</a>

%H Norihiro Nakashima, Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.

%F a(n) = Sum_{k=0..n} 3^(n-k)*(n!/k!)*binomial(n-1, k-1).

%F Recurrence: a(n) = (6*n-5)*a(n-1) - 9*(n-2)*(n-1)*a(n-2).

%F a(n) ~ n! * exp(2*sqrt(n/3) - 1/6) * 3^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Nov 21 2018

%p seq(coeff(series(factorial(n)*exp(x/(1-3*x)),x,n+1), x, n), n = 0 .. 17); # _Muniru A Asiru_, Nov 24 2018

%t a[n_] := Sum[3^(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] = (6n - 5)*a[n - 1] - 9(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* _Amiram Eldar_, Nov 19 2018 *)

%o (PARI) my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-3*x)))) \\ _Michel Marcus_, Nov 25 2018

%o (Magma) [1] cat [&+[3^(n-k)*Factorial(n) div Factorial(k)*Binomial(n-1, k-1): k in [0..n]]: n in [1.. 18]]; // _Vincenzo Librandi_, Dec 08 2018

%o (Sage) {c[1]:c[0]*factorial(c[1]) for c in (exp(x/(1-3*x))).taylor(x,0,25).coefficients()} # _G. C. Greubel_, Dec 14 2018

%o (GAP) Concatenation([1], List([1..25], n-> Sum([1..n], k-> 3^(n-k)*(Factorial(n)/Factorial(k))*Binomial(n-1, k-1)))); # _G. C. Greubel_, Dec 14 2018

%Y Cf. A000262, A025168, A321847, A321848, A321849, A321850 (analogs for k=1,2,4,5,6,7).

%K nonn

%O 0,3

%A _Ludovic Schwob_, Nov 19 2018