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A118395 Expansion of e.g.f. exp(x + x^3).

%I A118395 #47 Sep 08 2022 08:45:25
%S A118395 1,1,1,7,25,61,481,2731,10417,91225,681121,3493711,33597961,303321877,
%T A118395 1938378625,20282865331,211375647841,1607008257841,18157826367937,
%U A118395 212200671085975,1860991143630841,22560913203079021,289933758771407521,2869267483843753147,37116733726117707025
%N A118395 Expansion of e.g.f. exp(x + x^3).
%C A118395 Equals row sums of triangle A118394.
%H A118395 Robert Israel, <a href="/A118395/b118395.txt">Table of n, a(n) for n = 0..533</a>
%F A118395 E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) =  1 + (1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1) )), recursively defined continued fraction. - _Sergei N. Gladkovskii_, Feb 04 2013
%F A118395 a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp((n/3)^(1/3)-2*n/3). - _Vaclav Kotesovec_, Jun 02 2013
%F A118395 E.g.f.: A(x) = exp(x+x^3) satisfies A' - (1+3*x^2)*A = 0. - _Gheorghe Coserea_, Aug 24 2015
%F A118395 a(n+1) = a(n) + 3*n*(n-1)*a(n-2). - _Gheorghe Coserea_, Aug 24 2015
%F A118395 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-2*k)!. - _Seiichi Manyama_, Feb 25 2022
%p A118395 with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z, Z)))}, labeled], size=n)), n=0..22); # _Zerinvary Lajos_, Mar 18 2008
%t A118395 CoefficientList[Series[E^(x+x^3), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 02 2013 *)
%t A118395 T[n_, k_] := n!/(k!(n-3k)!);
%t A118395 a[n_] := Sum[T[n, k], {k, 0, Floor[n/3]}];
%t A118395 a /@ Range[0, 24] (* _Jean-François Alcover_, Nov 04 2020 *)
%o A118395 (PARI) a(n)=n!*polcoeff(exp(x+x^3+x*O(x^n)),n)
%o A118395 (PARI)
%o A118395 N=33;  x='x+O('x^N);
%o A118395 egf=exp(x+x^3);
%o A118395 Vec(serlaplace(egf))
%o A118395 /* _Joerg Arndt_, Sep 15 2012 */
%o A118395 (PARI) a(n) = n!*sum(k=0, n\3, binomial(n-2*k, k)/(n-2*k)!); \\ _Seiichi Manyama_, Feb 25 2022
%o A118395 (Magma) [n le 3 select 1 else Self(n-1) + 3*(n-2)*(n-3)*Self(n-3): n in [1..26]]; // _Vincenzo Librandi_, Aug 25 2015
%o A118395 (Sage)
%o A118395 def a(n):
%o A118395     if (n<3): return 1
%o A118395     else: return a(n-1) + 3*(n-1)*(n-2)*a(n-3)
%o A118395 [a(n) for n in (0..25)] # _G. C. Greubel_, Feb 18 2021
%Y A118395 Cf. A118394, A118396.
%Y A118395 Cf. A047974, A190875, A190877.
%K A118395 nonn
%O A118395 0,4
%A A118395 _Paul D. Hanna_, May 07 2006
%E A118395 Missing a(0)=1 prepended by _Joerg Arndt_, Sep 15 2012