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 A139383 Number of n-level labeled rooted trees with n leaves. 0
 1, 2, 12, 154, 3455, 120196, 5995892, 406005804, 35839643175, 3998289746065, 550054365477936, 91478394767427823, 18091315306315315610, 4196205472500769304318, 1128136777063831105273242 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1)), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. [From Gerald McGarvey, Aug 19 2009] LINKS FORMULA a(n)=T(n,n), T(n,m)=sum(i=1..n, stirling2(n,i)*T(i,m-1)), m>1, T(n,1)=1. [Vladimir Kruchinin, May 19 2012] EXAMPLE If we form a table from the family of sequences defined by: number of k-level labeled rooted trees with n leaves, then this sequence equals the diagonal in that table: n=1:A000012=[1,1,1,1,1,1,1,1,1,1,...]; n=2:A000110=[1,2,5,15,52,203,877,4140,21147,115975,...]; n=3:A000258=[1,3,12,60,358,2471,19302,167894,1606137,...]; n=4:A000307=[1,4,22,154,1304,12915,146115,1855570,26097835,...]; n=5:A000357=[1,5,35,315,3455,44590,660665,11035095,204904830,...]; n=6:A000405=[1,6,51,561,7556,120196,2201856,45592666,1051951026,...]; n=7:A001669=[1,7,70,910,14532,274778,5995892,148154860,4085619622,...]; n=8:A081624=[1,8,92,1380,25488,558426,14140722,406005804,13024655442,...]; n=9:A081629=[1,9,117,1989,41709,1038975,29947185,979687005,35839643175,..]. Row n in the above table equals column 0 of matrix power A008277^n where A008277 = triangle of Stirling numbers of 2nd kind: 1; 1,1; 1,3,1; 1,7,6,1; 1,15,25,10,1; 1,31,90,65,15,1; ... The name of this sequence is a generalization of the definition given in the above sequences by Christian G. Bower. PROG (PARI) {a(n)=local(E=exp(x+x*O(x^n))-1, F=x); for(i=1, n, F=subst(F, x, E)); n!*polcoeff(F, n)} (Maxima) T(n, m):=if m=1 then 1 else sum(stirling2(n, i)*T(i, m-1), i, 1, n); makelist(T(n, n), n, 1, 7); [Vladimir Kruchinin, May 19 2012] CROSSREFS Cf. A008277; A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629. Sequence in context: A085628 A177777 A053549 * A216351 A130529 A075631 Adjacent sequences:  A139380 A139381 A139382 * A139384 A139385 A139386 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 16 2008 STATUS approved

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