A156212 -id:A156212 - cp's OEIS frontend

A156216 G.f.: A(x) = exp( Sum_{n>=1} A000204(n)^n * x^n/n ), a power series in x with integer coefficients.

1, 1, 5, 26, 634, 32928, 5704263, 2470113915, 2978904483553, 9401949327631932, 79268874871208384494, 1762019469678472912173354, 103537245443913551792800303420, 16030602885085486700462431379649694

Offset: 0

Paul D. Hanna, Feb 06 2009

nonn

Compare to g.f. of Fibonacci sequence: exp( Sum_{n>=1} A000204(n)*x^n/n ), where A000204 is the Lucas numbers.

More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture).

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
log(A(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 +...

Cf. A000045, A000204, A156212.

Cf. A067961. [From Paul D. Hanna, Sep 13 2010]

{a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m+1)+fibonacci(m-1))^m*x^m/m)+x*O(x^n)),n)}

a(n) = (1/n)*Sum_{k=1..n} A000204(k)^k*a(n-k) for n>0, with a(0) = 1.

Logarithmic derivative forms A067961. [From Paul D. Hanna, Sep 13 2010]

A350520 The number of degree-n^2 polynomials over Z/2Z that can be written as f(f(x)) where f is a polynomial.

Original entry on oeis.org

1, 1, 3, 8, 14, 32, 60, 128, 248, 512, 1008, 2048, 4064, 8192, 16320, 32768, 65408, 131072, 261888, 524288, 1048064, 2097152, 4193280, 8388608, 16775168, 33554432, 67104768, 134217728, 268427264

Offset: 0

Peter Kagey, Jan 03 2022

			For n = 2, there are a(2) = 3 degree 4 polynomials of the form f(f(x)):
  x^4         = f(f(x)) when f(x) = x^2 or f(x) = x^2 + 1,
  x^4 + x     = f(f(x)) when f(x) = x^2 + x, and
  x^4 + x + 1 = f(f(x)) when f(x) = x^2 + x + 1.

Cf. A033991, A156212, A156232.

Conjecture:
  a(2n) = A033991(2^(n-1)) = 4^n - 2^(n-1) for n >= 1;
  a(2n+1) = 2^(2n+1) for n >= 1.
Conjecture from Hugo Pfoertner, Jan 09 2022: Terms starting at 3 coincide with {A156232}/8.

a(0) prepended and a(11)-a(28) from Martin Ehrenstein, Jan 14 2022

cp's OEIS Frontend

A156216 G.f.: A(x) = exp( Sum_{n>=1} A000204(n)^n * x^n/n ), a power series in x with integer coefficients.

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