This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A033185 Rooted tree triangle read by rows: a(n,k) = number of forests with n nodes and k rooted trees. 21
 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 7, 3, 1, 1, 48, 37, 18, 7, 3, 1, 1, 115, 96, 44, 19, 7, 3, 1, 1, 286, 239, 117, 46, 19, 7, 3, 1, 1, 719, 622, 299, 124, 47, 19, 7, 3, 1, 1, 1842, 1607, 793, 320, 126, 47, 19, 7, 3, 1, 1, 4766, 4235, 2095, 858, 327, 127, 47, 19, 7, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Leading column: A000081, rows sums: A000081 shifted. Also, number of multigraphs of k components, n nodes, and no cycles except one loop in each component. See link below to have a picture showing the bijection between rooted forests and multigraphs of this kind. - Washington Bomfim, Sep 04 2010 Number of rooted trees with n+1 nodes and degree of the root is k.- Michael Somos, Aug 20 2018 LINKS Alois P. Heinz, Rows n = 1..141, flattened R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 (2016), Table 2. FORMULA G.f.: 1/Product_{i>=1} (1-x*y^i)^A000081(i). - Vladeta Jovovic, Apr 28 2005 a(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000081(i)+Mi-1, Mi). - Washington Bomfim, May 12 2005 EXAMPLE Triangle begins:      1;      1,    1;      2,    1,   1;      4,    3,   1,   1;      9,    6,   3,   1,   1;     20,   16,   7,   3,   1,  1;     48,   37,  18,   7,   3,  1,  1;    115,   96,  44,  19,   7,  3,  1,  1;    286,  239, 117,  46,  19,  7,  3,  1,  1;    719,  622, 299, 124,  47, 19,  7,  3,  1,  1;   1842, 1607, 793, 320, 126, 47, 19,  7,  3,  1,  1; MAPLE with(numtheory): t:= proc(n) option remember; local d, j; `if` (n<=1, n,       (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))     end: b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *        binomial(t(i)+j-1, j), j=0..min(n/i, p)))))     end: a:= (n, k)-> b(n, n, k): seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012 MATHEMATICA nn=10; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0 == Series[f[x]-x Product[1/(1-x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; a[0]=0; g=Table[a[n], {n, 1, nn}]/.sol//Flatten; h[list_]:=Select[list, #>0&]; Map[h, Drop[CoefficientList[Series[x Product[1/(1-y x^i)^g[[i]], {i, 1, nn}], {x, 0, nn}], {x, y}], 2]]//Grid  (* Geoffrey Critzer, Nov 17 2012 *) t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_, k_] := b[n, n, k]; Table[a[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *) CROSSREFS Cf. A000081, A005197, A106240, A181360, A027852 (2nd column), A000226 (3rd column), A029855 (4th column). Sequence in context: A092056 A103574 A112682 * A217781 A204849 A105632 Adjacent sequences:  A033182 A033183 A033184 * A033186 A033187 A033188 KEYWORD nonn,tabl AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 21 15:03 EST 2018. Contains 317449 sequences. (Running on oeis4.)