A131518 - OEIS

%I #13 Jul 18 2022 20:42:42

%S 2,6,14,122,362,5282,20582,397154,2027090,46177922,303147902,

%T 7699478162,63517159994,1745540360930,17676592058582,517137940132802,

%U 6290714838241442,194139271606482434,2782486941099788270,90105513853333901042,1495993248737211995402,50671468195931300884322

%N Number of partitions of the graph G_n (defined below) into "strokes".

%C 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.

%C 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.

%H G. C. Greubel, <a href="/A131518/b131518.txt">Table of n, a(n) for n = 1..440</a>

%H R. J. Mathar, <a href="/A131518/a131518.pdf">Explanation of the Alekseyev formula</a>

%F 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

%e G_2 : o=o, two edges exist between v_1 and v_2.

%t f[n_, k_]:= If[EvenQ[n-k], Binomial[(n+k)/2, k], 0];

%t A088009[n_]:= n!*Sum[f[n-1, k-1]/k!, {k, 0, n}];

%t A131518[n_]:= If[EvenQ[n], 2*A088009[n] + n!*(n/2 +1), 2*A088009[n]];

%t Table[A131518[n], {n,1,30}] (* _G. C. Greubel_, Feb 14 2021 *)

%o (Sage)

%o def f(n, k): return binomial((n+k)/2, k) if (n-k)%2==0 else 0

%o def A088009(n): return factorial(n)*sum(f(n-1, k-1)/factorial(k) for k in (0..n))

%o def A131518(n): return 2*A088009(n) + (n/2 +1)*factorial(n) if (n%2==0) else 2*A088009(n)

%o [A131518(n) for n in (1..30)] # _G. C. Greubel_, Feb 14 2021

%Y Cf. A131520, A354228.

%K nonn

%O 1,1

%A _Yasutoshi Kohmoto_, Aug 15 2007, Oct 03 2007

%E More terms from _Max Alekseyev_, Sep 29 2007