A113070 -id:A113070 - cp's OEIS frontend

A384844 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Dorogovtsev-Goltsev-Mendes graph.

3, 9, 6, 27, 57, 21, 81, 351, 369, 60, 243, 1806, 3582, 1716, 156, 729, 8472, 26346, 24216, 6648, 384, 2187, 37683, 165375, 241032, 128880, 22896, 912, 6561, 162177, 938907, 1946676, 1670280, 584784, 72624, 2112, 19683, 683112, 4979928, 13697148, 16889340, 9580368, 2366256, 216768, 4800

Offset: 1

Andrew Howroyd, Jun 10 2025

			Triangle begins:
     3;
     9,     6;
    27,    57,     21;
    81,   351,    369,     60;
   243,  1806,   3582,   1716,    156;
   729,  8472,  26346,  24216,   6648,   384;
  2187, 37683, 165375, 241032, 128880, 22896, 912;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.

Main diagonal is A113070(n-1) for n > 1.
Column 1 is A000244.
Cf. A384843.

T(n)={ my(c=x^2*y/((1 - x)*(1 - 3*x + 2*(1 - y)*x^2)) + O(x*x^n), b=(1-2*x)*c/x, g = y*(1+b+2*c) + serconvol(b + c, b + c + y*c) + serconvol(y*c, b + 2*c)); [Vecrev(p/y)|p<-Vec(3*g/(1 - 3*x))]}
{ foreach(T(10), row, print(row)) }

T(n)={my(g=3*(1 - 2*(3 + y)*x + 3*(3 - y + y^2)*x^2 - 4*(1 - y)^2*x^3)/((1 - x)*(1 - 3*x)*(1 - (5 + 4*y)*x + 4*(1 - y)^2*x^2))); [Vecrev(p)|p<-Vec(g + O(x^n))]}
{ foreach(T(10), row, print(row)) }

G.f.: 3*x*y*(1 - 2*(3 + y)*x + 3*(3 - y + y^2)*x^2 - 4*(1 - y)^2*x^3)/((1 - x)*(1 - 3*x)*(1 - (5 + 4*y)*x + 4*(1 - y)^2*x^2)).

A384843(n) = Sum_{k=1..n} k*T(n,k).

A113072 Tridiagonal matrix associated with coordination sequences.

Original entry on oeis.org

1, 2, 2, 1, 4, 3, 0, 2, 6, 4, 0, 0, 3, 8, 5, 0, 0, 0, 4, 10, 6, 0, 0, 0, 0, 5, 12, 7, 0, 0, 0, 0, 0, 6, 14, 8, 0, 0, 0, 0, 0, 0, 7, 16, 9, 0, 0, 0, 0, 0, 0, 0, 8, 18, 10, 0, 0, 0, 0, 0, 0, 0, 0, 9, 20, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 22, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 24, 13, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Oct 14 2005

Coefficient array of expansion of ((1+x)/(1-r*x))^2, or self-convolution of coordination sequence for the infinite tree with valency r. Row sums are A008574, (coordination sequence for square lattice). Diagonal sums are A000027.

			Triangle begins
  1;
  2,  2;
  1,  4,  3;
  0,  2,  6,  4;
  0,  0,  3,  8,  5;
  0,  0,  0,  4, 10,  6;
  0,  0,  0,  0,  5, 12,  7;

Cf. A113070, A113071.

Column k has g.f. (1+x)^2*(k+1)x^k.

cp's OEIS Frontend

A384844 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Dorogovtsev-Goltsev-Mendes graph.

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