OFFSET
0,8
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened
FORMULA
T(n,0) = 1, T(n,k) = 0 if k<0 or n<k, else T(n,k) = Sum_{j=0..k} C(n-1,j) (A000272(j+1) T(n-j-1,k-j) + A057500(j+1) T(n-j-1,k-j-1)).
E.g.f.: exp(B(x,y)), where B(x,y) = Sum(Sum(A062734(n,k)*y^k*x^n/n!, k=0..n), n=1..infinity) = -1/2*log(1+LambertW(-x*y))+1/2*LambertW(-x*y) -1/4*LambertW(-x*y)^2-1/y *(LambertW(-x*y)+1/2 *LambertW(-x*y)^2). - Vladeta Jovovic, Sep 16 2008
EXAMPLE
T(4,4) = 15, because there are 15 simple graphs on 4 labeled nodes with 4 edges where each maximally connected subgraph has at most one cycle:
1-2 1-2 1-2 1-2 1-2 1-2 1 2 1 2 1-2 1 2 1 2 1-2 1-2 1-2 1 2
|/| |X |/ |\| X| \| |/| X| /| |\| |X |\ | | X |X|
4 3 4 3 4-3 4 3 4 3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4 3
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 3, 3, 1;
1, 6, 15, 20, 15;
1, 10, 45, 120, 210, 222;
...
MAPLE
cy:= proc(n) option remember; local t; binomial(n-1, 2) *add((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n, k) option remember; local j; if k=0 then 1 elif k<0 or n<k then 0 else add(binomial(n-1, j) *((j+1)^(j-1) *T(n-j-1, k-j) +cy(j+1) *T(n-j-1, k-j-1)), j=0..k) fi end: seq(seq(T(n, k), k=0..n), n=0..11);
MATHEMATICA
t[_, 0] = 1; t[n_, k_] /; (k<0 || n<k) = 0; t[n_, k_] := t[n, k] = Sum[Binomial[n-1, j]*(t[n-j-1, k-j]*(j+1)^(j-1) + 1/2*j!*Sum[1/((j+1)^m*(j-m+1)!), {m, 3, j+1}]*t[n-j-1, k-j-1]*(j+1)^(j+1)), {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Jan 15 2014, after Maple *)
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 15 2008
STATUS
approved