A035049 E.g.f. satisfies A(x) = x*(1+A(A(x))), A(0)=0.
1, 2, 12, 144, 2760, 74880, 2676240, 120234240, 6571393920, 426547296000, 32283270835200, 2808028566604800, 277433852555059200, 30836115140589158400, 3824551325912308992000, 525674251444773150720000, 79591811594194480508928000, 13205626859810397006618624000
Offset: 1
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..283
Programs
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Maple
A:= proc(n) option remember; `if`(n=0, 0, (T-> unapply( convert(series(x*(1+T(T(x))), x, n+1), polynom), x))(A(n-1))) end: a:= n-> coeff(A(n)(x), x, n)*n!: seq(a(n), n=1..20); # Alois P. Heinz, Aug 23 2008 # second Maple program: b:= proc(n, k) option remember; `if`(n=0, 1, add(k* a(j)*b(n-j, k-1)*binomial(n-1, j-1), j=1..n)) end: a:= n-> `if`(n=0, 1, b(n-1, n)): seq(a(n), n=1..20); # Alois P. Heinz, Aug 21 2019 -
Mathematica
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 *) -
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 */
Formula
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
Extensions
More terms from Alois P. Heinz, Aug 23 2008