A032173 Sequence (a(n): n >= 1) that shifts left 2 places under the "CHK" (necklace, identity, unlabeled) transform and has initial terms a(1) = a(2) = 1.

1, 1, 1, 1, 2, 3, 7, 12, 28, 55, 122, 258, 574, 1254, 2813, 6283, 14220, 32237, 73631, 168660, 388331, 896790, 2078822, 4832343, 11266422, 26332119, 61694574, 144858260, 340829231, 803427128, 1897269215, 4487725726

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Comments

From Petros Hadjicostas, Dec 29 2018: (Start)

a(n+2) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.

G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).

The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 15*x^4 + 36*x^5 + 81*x^6 + 197*x^7 + 455*x^8 + 1105*x^9 + 2618*x^10 + ... (The auxiliary sequence is given by sequence A322913.)

(End)

The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CHK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) - b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 2), and so on. - Petros Hadjicostas, Dec 31 2018

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• C. G. Bower, Transforms (2)

Crossrefs

Cf. A032171, A032174, A322913.

Programs

Mathematica

a[1] = a[2] = 1; c[1] = 1; c[2] = 3;
a[n_] := a[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
c[n_] := c[n] = n a[n] + Sum[c[s] a[n-s], {s, 1, n-1}];
Array[a, 32] (* Jean-François Alcover, Jan 02 2019 *)

PARI

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(p+O(x^n))} \\ Andrew Howroyd, Jun 20 2018

Extensions

Name modified by Petros Hadjicostas, Jan 01 2019