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

Author

Paul D. Hanna, Feb 06 2009

Keywords

• nonn

Comments

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).

Examples

			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 +...
		

Crossrefs

Cf. A000045, A000204, A156212.

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

Programs

PARI

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

Formula

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]