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A091695 Expansion of e.g.f. exp(x/(1-x)^3).

%I A091695 #20 Mar 07 2023 10:37:20
%S A091695 1,1,7,55,529,6121,82711,1273567,21945505,417540529,8680953511,
%T A091695 195582295591,4742407056817,123045795823705,3399348471640759,
%U A091695 99573135106176271,3081061456572152641,100382623544966098657,3433727597233037475655,123000248740384210119319,4603377404407810366309201
%N A091695 Expansion of e.g.f. exp(x/(1-x)^3).
%C A091695 Special values of the hypergeometric function 3F3: a(n) = n!*binomial(n+1,n-1) * hypergeom([ -n+1, 1/2*n+1, 1/2*n+3/2], [4/3, 5/3, 2], -4/27) for n>0.
%H A091695 Seiichi Manyama, <a href="/A091695/b091695.txt">Table of n, a(n) for n = 0..422</a>
%F A091695 E.g.f.: exp(x/(1-x)^3).
%F A091695 a(n) ~ 1/2*exp(-1/27-n^(1/4)*3^(3/4)/72+sqrt(3*n)/6+4/3*n^(3/4)*3^(1/4)-n)*3^(1/8)*n^(n-1/8). - _Vaclav Kotesovec_, Jun 27 2013
%F A091695 a(n) = n! * Sum_{k=0..n} binomial(n+2*k-1,n-k)/k!. - _Seiichi Manyama_, Mar 06 2023
%t A091695 CoefficientList[Series[E^(x/(1-x)^3), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o A091695 (PARI)
%o A091695 x='x+O('x^33);
%o A091695 Vec(serlaplace(exp( x/(1-x)^3 )))
%o A091695 /* _Joerg Arndt_, Sep 14 2012 */
%Y A091695 Column k=3 of A293012.
%Y A091695 Cf. A082579.
%K A091695 nonn
%O A091695 0,3
%A A091695 _Karol A. Penson_, Jan 29 2004
%E A091695 Prepended a(0)=1, _Joerg Arndt_, Sep 14 2012.