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

1,1

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.

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).

Example

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;
  ...

Prog

(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)) }

Crossrefs

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

Column 1 is A000244.

Cf. A384843.

Sequence in context: A224190 A223815 A275414 * A223309 A179483 A346108

Adjacent sequences: A384841 A384842 A384843 * A384845 A384846 A384847

Keyword

nonn,tabl

Author

Andrew Howroyd, Jun 10 2025

Status

approved