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E.g.f. satisfies A(x) = exp( x*A(x)/(1 - x * A(x)^3) ).
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%I #13 Aug 19 2023 06:28:58

%S 1,1,5,58,1061,26536,843457,32553424,1478813513,77304347776,

%T 4571222616701,301696674682624,21985118975444077,1753288356936334336,

%U 151887264799071753785,14203597499192539334656,1426051485043745729079953,153000280727938469281693696

%N E.g.f. satisfies A(x) = exp( x*A(x)/(1 - x * A(x)^3) ).

%H Michael De Vlieger, <a href="/A365013/b365013.txt">Table of n, a(n) for n = 0..341</a>

%F a(n) = n! * Sum_{k=0..n} (3*n-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.

%t Array[#!*Sum[ (3 # - 2 k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 18, 0] (* _Michael De Vlieger_, Aug 18 2023 *)

%o (PARI) a(n) = n!*sum(k=0, n, (3*n-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

%Y Cf. A052873, A365012.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 15 2023