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 A055277 Triangle T(n,k) of number of rooted trees with n nodes and k leaves, 1 <= k <= n. 38
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 18, 14, 5, 1, 0, 1, 12, 35, 39, 21, 6, 1, 0, 1, 16, 62, 97, 72, 30, 7, 1, 0, 1, 20, 103, 212, 214, 120, 40, 8, 1, 0, 1, 25, 161, 429, 563, 416, 185, 52, 9, 1, 0, 1, 30, 241, 804, 1344, 1268, 732, 270, 65, 10, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS Harary denotes the g.f. as P(x, y) on page 33 "... , and let P(x,y) = Sum Sum P_{nm} x^ny^m where P_{nm} is the number of planted trees with n points and m endpoints, in which again the plant has not been counted either as a point or as an endpoint." - Michael Somos, Nov 02 2014 REFERENCES F. Harary, Recent results on graphical enumeration, pp. 29-36 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 N. J. A. Sloane, Transforms Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.) FORMULA G.f. satisfies A(x, y) = x*y + x*EULER(A(x, y)) - x. Shifts up under EULER transform. G.f. satisfies A(x, y) = x*y - x + x * exp(Sum_{i>0} A(x^i, y^i) / i). [Harary, p. 34, equation (10)]. - Michael Somos, Nov 02 2014 Sum_k T(n, k) = A000081(n). - Michael Somos, Aug 24 2015 EXAMPLE From Joerg Arndt, Aug 18 2014: (Start) Triangle starts: 01: 1 02: 1    0 03: 1    1    0 04: 1    2    1    0 05: 1    4    3    1    0 06: 1    6    8    4    1    0 07: 1    9   18   14    5    1    0 08: 1   12   35   39   21    6    1    0 09: 1   16   62   97   72   30    7    1    0 10: 1   20  103  212  214  120   40    8    1    0 11: 1   25  161  429  563  416  185   52    9    1    0 12: 1   30  241  804 1344 1268  732  270   65   10    1    0 13: 1   36  348 1427 2958 3499 2544 1203  378   80   11    1    0 ... The trees with n=5 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are: : :     1:  [ 0 1 2 3 4 ]   1 :  O--o--o--o--o : :     2:  [ 0 1 2 3 3 ]   2 :  O--o--o--o :        .--o : :     3:  [ 0 1 2 3 2 ]   2 :  O--o--o--o :     .--o : :     4:  [ 0 1 2 3 1 ]   2 :  O--o--o--o :  .--o : :     5:  [ 0 1 2 2 2 ]   3 :  O--o--o :     .--o :     .--o : :     6:  [ 0 1 2 2 1 ]   3 :  O--o--o :     .--o :  .--o : :     7:  [ 0 1 2 1 2 ]   2 :  O--o--o :  .--o--o : :     8:  [ 0 1 2 1 1 ]   3 :  O--o--o :  .--o :  .--o : :     9:  [ 0 1 1 1 1 ]   4 :  O--o :  .--o :  .--o :  .--o : This gives [1, 4, 3, 1, 0], row n=5 of the triangle. (End) G.f. = x*(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*x^3 + y^4) + ...). MATHEMATICA rut[n_]:=rut[n]=If[n===1, {{}}, Join@@Function[c, Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]]; Table[Length[Select[rut[n], Count[#, {}, {-2}]===k&]], {n, 13}, {k, n}] (* Gus Wiseman, Mar 19 2018 *) PROG (PARI) {T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */ CROSSREFS Row sums give A000081. Columns 2 through 12: A002620(n-1), A055278-A055287. Cf. A055288, A055289, A055290. Cf. A001190, A003238, A004111, A055327, A214575, A290689, A298422, A298426, A301342, A301343, A301344, A301345. Sequence in context: A128307 A349394 A034369 * A301422 A055340 A058716 Adjacent sequences:  A055274 A055275 A055276 * A055278 A055279 A055280 KEYWORD nonn,tabl,eigen AUTHOR Christian G. Bower, May 09 2000 STATUS approved

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Last modified May 17 12:38 EDT 2022. Contains 353746 sequences. (Running on oeis4.)