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A035049 E.g.f. satisfies A(x) = x*(1+A(A(x))), A(0)=0.

%I A035049 #29 Jun 02 2024 11:08:20
%S A035049 1,2,12,144,2760,74880,2676240,120234240,6571393920,426547296000,
%T A035049 32283270835200,2808028566604800,277433852555059200,
%U A035049 30836115140589158400,3824551325912308992000,525674251444773150720000,79591811594194480508928000,13205626859810397006618624000
%N A035049 E.g.f. satisfies A(x) = x*(1+A(A(x))), A(0)=0.
%H A035049 Alois P. Heinz, <a href="/A035049/b035049.txt">Table of n, a(n) for n = 1..283</a>
%F A035049 a(n) = n!*T(n,1), T(n,m) = m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i) * C(i-1,k-1)*(-1)^i)*(-1)^k*C(n+k-1,n-1)), n>m, T(n,n)=1. - _Vladimir Kruchinin_, May 06 2012
%p A035049 A:= proc(n) option remember; `if`(n=0, 0, (T-> unapply(
%p A035049       convert(series(x*(1+T(T(x))), x, n+1), polynom), x))(A(n-1)))
%p A035049     end:
%p A035049 a:= n-> coeff(A(n)(x), x, n)*n!:
%p A035049 seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 23 2008
%p A035049 # second Maple program:
%p A035049 b:= proc(n, k) option remember; `if`(n=0, 1, add(k*
%p A035049       a(j)*b(n-j, k-1)*binomial(n-1, j-1), j=1..n))
%p A035049     end:
%p A035049 a:= n-> `if`(n=0, 1, b(n-1, n)):
%p A035049 seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 21 2019
%t A035049 T[n_, m_] := T[n, m] = If[n == m, 1, m/n*Sum[Sum[T[n-m, i]*Binomial[i-1, k-1]*(-1)^i, {i, k, n-m}]*(-1)^k*Binomial[n+k-1, n-1], {k, 1, n-m}]]; Table[n!*T[n, 1], {n, 1, 16}] (* _Jean-François Alcover_, Feb 12 2014, after _Vladimir Kruchinin_ *)
%o A035049 (Maxima) T(n,m):=if n=m then 1 else m/n*sum(sum(T(n-m,i)*binomial(i-1,k-1)*(-1)^i,i,k,n-m)*(-1)^k*binomial(n+k-1,n-1),k,1,n-m); makelist(n!*T(n,1),n,1,10); /* _Vladimir Kruchinin_, May 06 2012 */
%Y A035049 Cf. A001028, A030266.
%K A035049 nonn,eigen
%O A035049 1,2
%A A035049 _Christian G. Bower_, Oct 15 1998
%E A035049 More terms from _Alois P. Heinz_, Aug 23 2008