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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.

%I A384844 #12 Jun 14 2025 03:40:18
%S A384844 3,9,6,27,57,21,81,351,369,60,243,1806,3582,1716,156,729,8472,26346,
%T A384844 24216,6648,384,2187,37683,165375,241032,128880,22896,912,6561,162177,
%U A384844 938907,1946676,1670280,584784,72624,2112,19683,683112,4979928,13697148,16889340,9580368,2366256,216768,4800
%N 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.
%H A384844 Andrew Howroyd, <a href="/A384844/b384844.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%H A384844 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dorogovtsev-Goltsev-MendesGraph.html">Dorogovtsev-Goltsev-Mendes Graph</a>.
%F A384844 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)).
%F A384844 A384843(n) = Sum_{k=1..n} k*T(n,k).
%e A384844 Triangle begins:
%e A384844      3;
%e A384844      9,     6;
%e A384844     27,    57,     21;
%e A384844     81,   351,    369,     60;
%e A384844    243,  1806,   3582,   1716,    156;
%e A384844    729,  8472,  26346,  24216,   6648,   384;
%e A384844   2187, 37683, 165375, 241032, 128880, 22896, 912;
%e A384844   ...
%o A384844 (PARI)
%o A384844 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))]}
%o A384844 { foreach(T(10), row, print(row)) }
%o A384844 (PARI)
%o A384844 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))]}
%o A384844 { foreach(T(10), row, print(row)) }
%Y A384844 Main diagonal is A113070(n-1) for n > 1.
%Y A384844 Column 1 is A000244.
%Y A384844 Cf. A384843.
%K A384844 nonn,tabl
%O A384844 1,1
%A A384844 _Andrew Howroyd_, Jun 10 2025