A067145 Shifts left under reversion.

1, 1, -1, 3, -13, 69, -419, 2809, -20353, 157199, -1281993, 10963825, -97828031, 907177801, -8716049417, 86553001779, -886573220093, 9351927111901, -101447092428243, 1130357986741545, -12923637003161409, 151479552582252239, -1818756036793636033

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..200

N. J. A. Sloane, Transforms

Formula

G.f. satisfies A^(-1)(x) = A(x)/x - 1.

G.f. satisfies: A(A(x)) = (1+x)*A(x) = g.f. of A107094. - Paul D. Hanna, May 12 2005

G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(a0,a1,a2) = a1 - a2 + a0*a1. - Michael Somos, May 21 2005

a(n) = T(n-1,1), n > 1, a(1) = 1, T(n,m) = (m/n) * Sum_{k=1..n-m} T(n-m,k) * (-1)^k * binomial(k+n-1, n-1), n > m, T(n,n) = 1. - Vladimir Kruchinin, May 06 2012

Mathematica

Nest[InverseSeries[#] x + x &, x + O[x]^2, 50][[3]] (* Vladimir Reshetnikov, Aug 07 2019 *)

Prog

(PARI) {a(n)=local(A); if(n<1, 0, A=x+O(x^2); for(i=2, n, A=x*(1+serreverse(A))); polcoeff(A, n))} /* Michael Somos, May 21 2005 */
(Maxima) T(n, m):=if n=m then 1 else m/n*sum(T(n-m, k)*(-1)^k*binomial(k+n-1, n-1), k, 1, n-m); a(n):=if n=1 then 1 else T(n-1, 1); [Vladimir Kruchinin, May 06 2012]

Crossrefs

Cf. A107094.

Apart from signs, same as A088714. - Philippe Deléham, Jun 18 2006

Cf. A309254, A309564.

Sequence in context: A368708 A352855 A088714 * A192739 A088368 A196794

Adjacent sequences: A067142 A067143 A067144 * A067146 A067147 A067148

Keyword

sign,eigen

Author

Christian G. Bower, Jan 03 2002

Status

approved