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 A112739 Array counting nodes in rooted trees of height n in which the root and internal nodes have valency k (and the leaf nodes have valency one). 10
 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 2, 1, 1, 5, 10, 7, 2, 1, 1, 6, 17, 22, 9, 2, 1, 1, 7, 26, 53, 46, 11, 2, 1, 1, 8, 37, 106, 161, 94, 13, 2, 1, 1, 9, 50, 187, 426, 485, 190, 15, 2, 1, 1, 10, 65, 302, 937, 1706, 1457, 382, 17, 2, 1, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 19 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Rows of the square array have g.f. (1+x)/((1-x)(1-kx)). They are the partial sums of the coordination sequences for the infinite tree of valency k. Row sums are A112740. Rows of the square array are successively: A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023. - Philippe Deléham, Feb 22 2014 REFERENCES L. He, X. Liu and G. Strang, (2003) Trees with Cantor Eigenvalue Distribution. Studies in Applied Mathematics 110 (2), 123-138. L. He, X. Liu and G. Strang, Laplacian eigenvalues of growing trees, Proc. Conf. on Math. Theory of Networks and Systems, Perpignan (2000). LINKS FORMULA As a square array read by antidiagonals, T(n, k)=sum{j=0..k, (2-0^j)*(n-1)^(k-j)}; T(n, k)=(n(n-1)^k-2)/(n-2), n<>2, T(2, n)=2n+1; T(n, k)=sum{j=0..k, (n(n-1)^j-0^j)/(n-1)}, j<>1. As a triangle read by rows, T(n, k)=if(k<=n, sum{j=0..k, (2-0^j)*(n-k-1)^(k-j)}, 0). EXAMPLE As a square array, rows begin 1,1,1,1,1,1,... (A000012) 1,2,2,2,2,2,... (A040000) 1,3,5,7,9,11,... (A005408) 1,4,10,22,46,94,... (A033484) 1,5,17,53,161,485,... (A048473) 1,6,26,106,426,1706,... (A020989) 1,7,37,187,937,4687,... (A057651) 1,8,50,302,1814,10886,... (A061801) As a number triangle, rows start 1; 1,1; 1,2,1; 1,3,2,1; 1,4,5,2,1; 1,5,10,7,2,1; CROSSREFS Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023. Sequence in context: A093628 A186807 A114282 * A308813 A225640 A194543 Adjacent sequences:  A112736 A112737 A112738 * A112740 A112741 A112742 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Sep 16 2005 STATUS approved

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Last modified October 17 16:51 EDT 2019. Contains 328120 sequences. (Running on oeis4.)