A184818 E.g.f.: A(x) = Sum_{n>=0} (-log(1-x))^[n*phi] / [n*phi]!, where [n*phi] = A000201(n), the lower Wythoff sequence, and phi = (1+sqrt(5))/2.

1, 1, 1, 3, 13, 69, 431, 3100, 25264, 230301, 2323448, 25713402, 309822547, 4038325082, 56625410687, 850040474751, 13603082015860, 231189547428654, 4158861518106668, 78949554006168724, 1577308905369288069

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: A(x) = 1/(1-x) - Sum_{n>=1} (-log(1-x))^[n*phi^2] / [n*phi^2]!, where [n*phi^2] = A001950(n), the upper Wythoff sequence.

a(n) = n! - A184819(n) for n>0.

Example

 E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 13*x^4/4! + 69*x^5/5! +...
The series expansion begins:
A(x) = 1 - log(1-x) - log(1-x)^3/3! + log(1-x)^4/4! + log(1-x)^6/6! + log(1-x)^8/8! - log(1-x)^9/9! +...+ (-log(1-x))^A000201(n)/A000201(n)! +...
The complementary series begins:
A(x) = 1/(1-x) - log(1-x)^2/2! + log(1-x)^5/5! + log(1-x)^7/7! - log(1-x)^10/10! + log(1-x)^13/13! +...+ -(-log(1-x))^A001950(n)/A001950(n)! +...

Prog

 (PARI) {a(n)=local(phi=(sqrt(5)+1)/2, A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(k=1, n, (-log(1-x+x*O(x^n)))^floor(k*phi)/floor(k*phi)!+x*O(x^n))); n!*polcoeff(A, n)}

Crossrefs

Cf. A184819, A000201, A001950.

Sequence in context: A192739 A088368 A196794 * A352370 A007808 A104989

Adjacent sequences: A184815 A184816 A184817 * A184819 A184820 A184821

Keyword

nonn

Author

Paul D. Hanna, Jan 22 2011

Status

approved