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

Author

Andrew Howroyd, Jun 10 2025

Keywords

• nonn
• tabl

Examples

			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;
  ...
		

Links

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

Crossrefs

Main diagonal is A113070(n-1) for n > 1.

Column 1 is A000244.

Cf. A384843.

Programs

PARI

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)) }

PARI

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)) }

Formula

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).