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A346258 E.g.f.: exp(x) / (1 - 3 * x)^(1/3).

%I A346258 #18 Aug 25 2025 17:31:40
%S A346258 1,2,7,44,421,5366,84907,1601552,35052649,872931626,24368595631,
%T A346258 753607111412,25572085243597,944609383245854,37731673388579731,
%U A346258 1620520035001182296,74466516342569480017,3645540855448417250642,189415873005295070803159,10410429682102309433442236
%N A346258 E.g.f.: exp(x) / (1 - 3 * x)^(1/3).
%C A346258 Binomial transform of A007559.
%F A346258 a(n) = Sum_{k=0..n} binomial(n,k) * A007559(k).
%F A346258 a(n) ~ n! * exp(1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Aug 14 2021
%F A346258 a(n+2) = (3*n+5)*a(n+1)-3*(n+1)*a(n). - _Tani Akinari_, Sep 08 2023
%p A346258 g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
%p A346258 a:= n-> add(binomial(n, k)*g(k), k=0..n):
%p A346258 seq(a(n), n=0..19);  # _Alois P. Heinz_, Aug 10 2021
%t A346258 nmax = 19; CoefficientList[Series[Exp[x]/(1 - 3 x)^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
%t A346258 Table[Sum[Binomial[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]
%t A346258 Table[HypergeometricU[1/3, n + 4/3, 1/3]/3^(1/3), {n, 0, 19}]
%t A346258 With[{nn=20},CoefficientList[Series[Exp[x]/CubeRoot[1-3x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 25 2025 *)
%o A346258 (Maxima) a[n]:=if n<2 then n+1 else (3*n-1)*a[n-1]+3*(1-n)*a[n-2];
%o A346258 makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 08 2023 */
%Y A346258 Cf. A000522, A007559, A010845, A033030, A084262, A347012, A347013, A347014.
%K A346258 nonn,changed
%O A346258 0,2
%A A346258 _Ilya Gutkovskiy_, Aug 10 2021