A035049 - OEIS

%I #29 Jun 02 2024 11:08:20

%S 1,2,12,144,2760,74880,2676240,120234240,6571393920,426547296000,

%T 32283270835200,2808028566604800,277433852555059200,

%U 30836115140589158400,3824551325912308992000,525674251444773150720000,79591811594194480508928000,13205626859810397006618624000

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

%H Alois P. Heinz, <a href="/A035049/b035049.txt">Table of n, a(n) for n = 1..283</a>

%F 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 A:= proc(n) option remember; `if`(n=0, 0, (T-> unapply(

%p       convert(series(x*(1+T(T(x))), x, n+1), polynom), x))(A(n-1)))

%p     end:

%p a:= n-> coeff(A(n)(x), x, n)*n!:

%p seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 23 2008

%p # second Maple program:

%p b:= proc(n, k) option remember; `if`(n=0, 1, add(k*

%p       a(j)*b(n-j, k-1)*binomial(n-1, j-1), j=1..n))

%p     end:

%p a:= n-> `if`(n=0, 1, b(n-1, n)):

%p seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 21 2019

%t 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 (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 Cf. A001028, A030266.

%K nonn,eigen

%O 1,2

%A _Christian G. Bower_, Oct 15 1998

%E More terms from _Alois P. Heinz_, Aug 23 2008