OFFSET
0,3
COMMENTS
See the array and triangle A199571 for the general cycle graph C_N counting.
With offset = 1: Number of ways to separate n distinguishable objects into an odd size pile and an even size pile. For example: a(3) = 4 because we have: {{1},{2,3}}; {{2},{1,3}}; {{3},{1,2}}; {{1,2,3},{}}. - Geoffrey Critzer, Jun 10 2013
Inverse Stirling transform of A065143. - Vladimir Reshetnikov, Nov 01 2015
LINKS
R. J. Mathar, Counting Walks on Finite Graphs, Section 1.
Index entries for linear recurrences with constant coefficients, signature (0,4).
FORMULA
a(n) = (2^n + (-2)^n)/2 = 2^(n-1)*(1 + (-1)^n).
O.g.f.: 1/(1-(2*x)^2).
E.g.f.: cosh(2*x)=U(0) where U(k) = 1 + 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 + (4*k+3)*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 23 2012
EXAMPLE
a(2) = 4 from starting with vertex no. 1, with edges e1 and e2 to vertex no. 2: e1e1, e2e2, e1e2 and e2e1.
MATHEMATICA
nn = 39; Drop[Range[0, nn]! CoefficientList[Series[ Sinh[x] Cosh[x], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jun 10 2013 *)
PROG
(PARI) vector(100, n, n--; (2^(n) +(-2)^n)/2) \\ Altug Alkan, Nov 02 2015
CROSSREFS
KEYWORD
nonn,easy,walk,changed
AUTHOR
Wolfdieter Lang, Nov 08 2011
STATUS
approved