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 A144163 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph is either a tree or a cycle. 3
 1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 3, 1, 10, 45, 120, 150, 12, 1, 15, 105, 455, 1185, 1473, 70, 1, 21, 210, 1330, 5565, 14469, 18424, 465, 1, 28, 378, 3276, 19635, 81060, 213990, 280200, 3507, 1, 36, 630, 7140, 57393, 334656, 1385076, 3732300, 5029218, 30016 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA T(n,k) = A138464(n,k) + Sum_{j=3..k} C(n,j) A138464(n-j,k-j) A144161 (j,j). EXAMPLE T(4,3) = 20, because there are 20 simple graphs on 4 labeled nodes with 3 edges, where each maximally connected subgraph is either a tree or a cycle, 16 of these graphs consist of a single tree with 4 nodes and 4 consist of a cycle with 3 and a tree with 1 node:   .1-2. .1-2. .1.2. .1.2. .1-2. .1-2. .1-2. .1-2. .1-2. .1.2.   .|\.. ../|. ..\|. .|/.. .|... ...|. ../.. ..\.. .|.|. .|.|.   .4.3. .4.3. .4-3. .4-3. .4-3. .4-3. .4-3. .4-3. .4.3. .4-3.   .   .1.2. .1.2. .1-2. .1.2. .1.2. .1.2. .1.2. .1.2. .1-2. .1-2.   .|/|. .|\|. ..X.. ..X|. ..X.. .|X.. ../|. .|\.. .|/.. ..\|.   .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,   3;   1, 10, 45, 120, 150, 12; MAPLE f:= proc(n, k) option remember; local j; if k=0 then 1 elif k<0 or n<=k then 0 elif k=n-1 then n^(n-2) else add(binomial(n-1, j) *f(j+1, j) *f(n-1-j, k-j), j=0..k) fi end: c:= proc(n, k) option remember; local i, j; if k=0 then 1 elif k<0 or n

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Last modified August 7 19:57 EDT 2020. Contains 336279 sequences. (Running on oeis4.)