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A193277 Triangle T(n,k), n>=1, 0<=k<=(3+3^n)/2, read by rows: row n gives the coefficients of the chromatic polynomial of the Sierpinski gasket graph S_n, highest powers first.

%I A193277 #25 Jun 16 2025 15:30:47
%S A193277 1,-3,2,0,1,-9,32,-56,48,-16,0,1,-27,339,-2625,14016,-54647,160663,
%T A193277 -362460,631828,-848736,866640,-653248,343744,-112896,17408,0,1,-81,
%U A193277 3204,-82476,1553454,-22823259,272286183,-2711405961,22990179324
%N A193277 Triangle T(n,k), n>=1, 0<=k<=(3+3^n)/2, read by rows: row n gives the coefficients of the chromatic polynomial of the Sierpinski gasket graph S_n, highest powers first.
%C A193277 The Sierpinski graph S_n has (3+3^n)/2 vertices and 3^n edges. The chromatic polynomial of S_n has (3+3^n)/2+1 coefficients.
%H A193277 Alois P. Heinz, <a href="/A193277/b193277.txt">Rows n = 1..7, flattened</a>
%H A193277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>.
%H A193277 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>.
%H A193277 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>.
%e A193277 3 example graphs:                        o
%e A193277 .                                       / \
%e A193277 .                                      o---o
%e A193277 .                                     / \ / \
%e A193277 .                       o            o---o---o
%e A193277 .                      / \          / \     / \
%e A193277 .            o        o---o        o---o   o---o
%e A193277 .           / \      / \ / \      / \ / \ / \ / \
%e A193277 .          o---o    o---o---o    o---o---o---o---o
%e A193277 Graph:      S_1        S_2              S_3
%e A193277 Vertices:    3          6                15
%e A193277 Edges:       3          9                27
%e A193277 The Sierpinski graph S_1 is equal to the cycle graph C_3 with chromatic polynomial q^3 -3*q^2 +2*q => [1, -3, 2, 0].
%e A193277 Triangle T(n,k) begins:
%e A193277 1,    -3,       2,           0;
%e A193277 1,    -9,      32,         -56,           48,              -16,  ...
%e A193277 1,   -27,     339,       -2625,        14016,           -54647,  ...
%e A193277 1,   -81,    3204,      -82476,      1553454,        -22823259,  ...
%e A193277 1,  -243,   29295,    -2336013,    138604878,      -6526886841,  ...
%e A193277 1,  -729,  265032,   -64069056,  11585834028,   -1671710903793,  ...
%e A193277 1, -2187, 2389419, -1738877625, 948268049436, -413339609377179,  ...
%Y A193277 Cf. A000244, A067771, A185442, A193233.
%K A193277 sign,tabf,look,hard
%O A193277 1,2
%A A193277 _Alois P. Heinz_, Jul 20 2011