A081875 a(n) = Sum_{d|n} phi(n/d)*C(2*d,d)/2.
1, 4, 12, 40, 130, 480, 1722, 6480, 24336, 92520, 352726, 1352640, 5200312, 20060040, 77559060, 300546720, 1166803126, 4537592928, 17672631918, 68923357200, 269128940724, 1052049834616, 4116715363822, 16123803207552
Offset: 1
Examples
G.f. = x + 4*x^2 + 12*x^3 + 40*x^4 + 130*x^5 + 480*x^6 + 1722*x^7 + ...
Links
- Robert Israel, Table of n, a(n) for n = 1..1656
- V. A. Liskovets, Enumerative formulas for unrooted planar maps: a pattern, Electron. J. Combin., 11:1 (2004), R88.
- J. Malenfant, On the Matrix-Element Expansion of a Circulant Determinant, arXiv preprint arXiv:1502.06012 [math.NT], 2015.
Programs
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Maple
f:= proc(n) local d; add(numtheory:-phi(n/d)*binomial(2*d,d)/2, d = numtheory:-divisors(n)) end proc: map(f, [$1..30]); # Robert Israel, Nov 29 2024
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Mathematica
Table[Fold[ #1+EulerPhi[n/#2]*Binomial[2#2, #2]/2&, 0, Divisors[n]], {n, 1, 32}] a[ n_] := If[ n < 0, 0, Sum[ Binomial[2 d, d] EulerPhi[n / d], {d, Divisors @ n}] / 2]; (* Michael Somos, Nov 01 2014 *) -
PARI
{a(n) = if( n<1, 0, sumdiv(n, d, binomial(2*d, d) * eulerphi(n/d)) / 2)}; /* Michael Somos, Nov 01 2014 */
Formula
a(n) ~ 2^(2*n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 08 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} C(2*gcd(n,k),gcd(n,k))/2.
a(n) = Sum_{k=1..n} A088218(gcd(n,k)). (End)