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A289722 Irregular triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Apollonian network.

%I A289722 #17 Feb 16 2025 08:33:49
%S A289722 6,15,6,42,72,6,123,522,258,366,2970,3894,396,1095,14838,37332,13680,
%T A289722 216,3282,68736,278490,224928,24624,9843,303918,1779678,2517228,
%U A289722 754704,22032,29526,1303938,10269150,22233096,13114656,1489104,7776
%N A289722 Irregular triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the n-Apollonian network.
%C A289722 Row n has k in the range 1..1+floor(2*n/3).
%C A289722 Table gives the coefficients of the Wiener polynomials. The degree of the polynomial corresponds to the diameter of the graph.
%H A289722 Andrew Howroyd, <a href="/A289722/b289722.txt">Table of n, a(n) for n = 1..883</a>
%H A289722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>
%H A289722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%e A289722 Triangle begins:
%e A289722 6;
%e A289722 15, 6;
%e A289722 42, 72, 6;
%e A289722 123, 522, 258;
%e A289722 366, 2970, 3894, 396;
%e A289722 1095, 14838, 37332, 13680, 216;
%e A289722 3282, 68736, 278490, 224928, 24624;
%e A289722 9843, 303918, 1779678, 2517228, 754704, 22032;
%e A289722 29526, 1303938, 10269150, 22233096, 13114656, 1489104, 7776;
%e A289722 ...
%t A289722 R[dp_, peq_, p1_, p2_] := {3*(dp - x + peq^2 + (2 + 7*x)*p1^2 + (7 + 2*x)*p2^2 + (4 + 2*x)*peq*p1 + 6*peq*p2 + 2*(4 + 5*x)*p1*p2 + x*(peq + 3*p1 + 3*p2)), x*(1 + 3*p1), 2*(p1 + p2), peq + p2};
%t A289722 A[n_] := (v = {6*x, x, 0, 0}; For[i = 2, i <= n, i++, v = R @@ v]; v[[1]]);
%t A289722 Table[CoefficientList[A[n], x] // Rest, {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Dec 28 2017, after _Andrew Howroyd_ *)
%o A289722 (PARI)
%o A289722 R(dp, peq,p1,p2,x) = {[3*(dp - x + peq^2 + (2+7*x)*p1^2 + (7+2*x)*p2^2 + (4+2*x)*peq*p1 + 6*peq*p2 + 2*(4+5*x)*p1*p2 + x*(peq+3*p1+3*p2)), x*(1+3*p1), 2*(p1+p2), peq+p2]}
%o A289722 A(n,x) = {my(v=[6*x,x,0,0,x]); for(i=2, n, v=R(v[1],v[2],v[3],v[4],x)); v[1]}
%o A289722 for (n=1,10,print(Vec(polrecip(A(n,x))),";" ))
%Y A289722 Cf. A289022.
%K A289722 nonn,tabf
%O A289722 1,1
%A A289722 _Andrew Howroyd_, Sep 02 2017