proposed

approved

editing

proposed

Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, A238276, A138894, A090843, A199023.

easy,nonn,changed,tabl

approved

editing

editing

approved

Cf. A112468, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801.

proposed

editing

editing

proposed

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

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)

Cf. A112468

approved

editing

_Paul Barry (pbarry(AT)wit.ie), _, Sep 16 2005

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).

easy,nonn,new

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).

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

0,5

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.

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).

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).

As a square array, rows begin

1,1,1,1,1,1,...

1,2,2,2,2,2,...

1,3,5,7,9,11,...

1,4,10,22,46,94,...

1,5,17,53,161,485,...

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;

easy,nonn,new

Paul Barry (pbarry(AT)wit.ie), Sep 16 2005

approved