OFFSET
1,4
COMMENTS
Proof of the formula: check that the associated combinatorial Laplacian has eigenvalues {0,..n-1}\ {floor((n+1)/2)} by exhibiting a basis of eigenvectors (which are very simple).
REFERENCES
N. Biggs, Algebraic Graph Theory, Cambridge University Press (1974).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..450
Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
Pierre-Alain Sallard, Coefficients of repeated integrals of hyperbolic cosine.
FORMULA
a(n) = (n-1)!/floor((n+1)/2).
a(n+1) = n!/floor(n/2 + 1). - M. F. Hasler, Apr 21 2015
1/a(n+1) is the coefficient of the power series of 3*exp(x)/4 + 1/4*exp(-x) + x/2*exp(x) ; this function is the sum of f_n(x) where f_0(x)=cosh(x) and f_{n+1} is the primitive of f_n. - Pierre-Alain Sallard, Dec 15 2018
EXAMPLE
a(1)=a(2)=a(3)=1 because the corresponding graphs are trees.
a(4)=3 because the corresponding graph is a triangle with one of its vertices adjacent to a fourth vertex.
MAPLE
a:=n->(n-1)!/floor((n+1)/2);
MATHEMATICA
Function[x, 1/x] /@
CoefficientList[Series[3*Exp[x]/4 + 1/4*Exp[-x] + x/2*Exp[x], {x, 0, 10}], x] (* Pierre-Alain Sallard, Dec 15 2018 *)
Table[(n - 1)! / Floor[(n + 1) / 2], {n, 1, 30}] (* Vincenzo Librandi, Dec 15 2018 *)
PROG
(PARI) A107991(n)=(n-1)!/round(n/2) \\ M. F. Hasler, Apr 21 2015
(Magma) [Factorial(n-1)/Floor((n+1)/2): n in [1..25]]; // Vincenzo Librandi, Dec 15 2018
(GAP) List([1..20], n->Factorial(n-1)/Int((n+1)/2)); # Muniru A Asiru, Dec 15 2018
(SageMath) [factorial(n-1)/floor((n+1)/2) for n in range(1, 24)] # Stefano Spezia, May 10 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Roland Bacher, Jun 13 2005
STATUS
approved