A309509 G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).

0, 1, 2, 2, 2, 2, 0, 4, 6, -58, 100, 1052, -5924, -21972, 322020, 332392, -21168682, 29068598, 1724404180, -7070346036, -172304798980, 1290100381724, 20728501384592, -247269172883976, -2936888518668676, 53037176259027580, 477640220538178184

Offset: 0

Author

Vladimir Reshetnikov, Aug 05 2019

Keywords

• sign

Comments

A(x) is sometimes called a functional square root, or half-iterate of F(x).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..488
• Gottfried Helms, Coefficients for fractional iterates exp(x)-1.
• Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.
• Wikipedia, Functional square root.

Crossrefs

Cf. A001787, A027436, A030274, A030275, A184011.

Programs

Mathematica

half[q_] := half[q] = Module[{h}, h[0] = 0; h[1] = 1; h[n_Integer] := h[n] = Module[{c}, c[m_Integer /; m < n] := h[m]; c[n] /. Solve[q[n] == Sum[k! c[k] BellY[n, k, Table[m! c[m], {m, n - k + 1}]], {k, n}]/n!, c[n]][[1]]]; h]; a[n_Integer] := a[n] = half[Function[k, k 2^(k-1)]][n]; Table[a[n], {n, 0, 26}]

Formula

Define the sequence b(n,k) as follows. If nSeiichi Manyama, May 03 2024