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A131518
Number of partitions of the graph G_n (defined below) into "strokes".
6
2, 6, 14, 122, 362, 5282, 20582, 397154, 2027090, 46177922, 303147902, 7699478162, 63517159994, 1745540360930, 17676592058582, 517137940132802, 6290714838241442, 194139271606482434, 2782486941099788270, 90105513853333901042, 1495993248737211995402, 50671468195931300884322
OFFSET
1,1
COMMENTS
Here G_n = {V_n, E_n}, V_n = {v_1, v_2}, E_n = {e_1, e_2, ..., e_n}; for all i, e_i = v_1v_2.
Given an undirected graph G=(V,E), its partition into strokes is a collection of directed edge-disjoint paths (viewed as sets of directed edges) on V such that (i) union of any two paths is not a path; (ii) union of corresponding undirected paths is E.
FORMULA
For odd n, a(n)=2*A088009(n); for even n, a(n)=2*A088009(n)+n!*(n/2+1). The first term stands for partitions with paths starting and ending in different vertices. The second term (that exists only for even n) stands for partitions with paths starting and ending at the same vertex (there are at most 2 such paths starting and ending in v_1 and v_2 respectively, each path consists of even number of edges). - Max Alekseyev, Sep 29 2007
EXAMPLE
G_2 : o=o, two edges exist between v_1 and v_2.
MATHEMATICA
f[n_, k_]:= If[EvenQ[n-k], Binomial[(n+k)/2, k], 0];
A088009[n_]:= n!*Sum[f[n-1, k-1]/k!, {k, 0, n}];
A131518[n_]:= If[EvenQ[n], 2*A088009[n] + n!*(n/2 +1), 2*A088009[n]];
Table[A131518[n], {n, 1, 30}] (* G. C. Greubel, Feb 14 2021 *)
PROG
(Sage)
def f(n, k): return binomial((n+k)/2, k) if (n-k)%2==0 else 0
def A088009(n): return factorial(n)*sum(f(n-1, k-1)/factorial(k) for k in (0..n))
def A131518(n): return 2*A088009(n) + (n/2 +1)*factorial(n) if (n%2==0) else 2*A088009(n)
[A131518(n) for n in (1..30)] # G. C. Greubel, Feb 14 2021
CROSSREFS
Sequence in context: A308568 A296054 A333121 * A222201 A130642 A133933
KEYWORD
nonn
AUTHOR
Yasutoshi Kohmoto, Aug 15 2007, Oct 03 2007
EXTENSIONS
More terms from Max Alekseyev, Sep 29 2007
STATUS
approved