cp's OEIS Frontend

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.

%I A184818 #8 Jun 30 2025 21:09:42
%S A184818 1,1,1,3,13,69,431,3100,25264,230301,2323448,25713402,309822547,
%T A184818 4038325082,56625410687,850040474751,13603082015860,231189547428654,
%U A184818 4158861518106668,78949554006168724,1577308905369288069
%N 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.
%F A184818 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.
%F A184818 a(n) = n! - A184819(n) for n>0.
%e A184818 E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 13*x^4/4! + 69*x^5/5! +...
%e A184818 The series expansion begins:
%e A184818 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)! +...
%e A184818 The complementary series begins:
%e A184818 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)! +...
%o A184818 (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)}
%Y A184818 Cf. A184819, A000201, A001950.
%K A184818 nonn
%O A184818 0,4
%A A184818 _Paul D. Hanna_, Jan 22 2011