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Triangle read by rows: CP(n,i) for n>=0 and 3n+1 >= i >= 0, gives the absolute value of the coefficients of the chromatic polynomial of C_3 X P_(n+1) factored in the form x(x-1)^i.
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%I #17 Feb 14 2021 13:03:11

%S 1,1,1,4,8,9,4,1,7,25,57,87,89,56,16,1,10,51,171,411,735,986,977,684,

%T 304,64,1,13,86,378,1219,3027,5930,9254,11485,11185,8304,4448,1536,

%U 256,1,16,130,705,2835,8918,22618,47055,81005,115630,136300,131225,101140

%N Triangle read by rows: CP(n,i) for n>=0 and 3n+1 >= i >= 0, gives the absolute value of the coefficients of the chromatic polynomial of C_3 X P_(n+1) factored in the form x(x-1)^i.

%H Alois P. Heinz, <a href="/A123531/b123531.txt">Rows n = 0..81, flattened</a>

%H T. Pfaff and J. Walker, <a href="https://projecteuclid.org/euclid.mjms/1316032776">The Chromatic Polynomial of P_2 X P_n and C_3 X P_n</a>, Missouri J. Math. Sci., 20, Issue 3 (2008), 169-177.

%F CP(n,i) = CP(n-1,i) +3*CP(n-1,i-1) +5*CP(n-1,i-2) +4*CP(n-1,i-3), with CP(0,0) = CP(0,1) = 1; n>=0 and 3n+1 >= i >= 0.

%e The chromatic polynomial of C_3 X P_2 is: x(x-1)^5 -4*x(x-1)^4 +8*x(x-1)^3 -9*x(x-1)^2 +4*x(x-1)^1 and so CP(1,0) = 1, CP(1,1) = 4, CP(1,2) = 8, CP(1,3) = 9 and CP(1,4) = 4.

%e Triangle begins:

%e 1, 1;

%e 1, 4, 8, 9, 4;

%e 1, 7, 25, 57, 87, 89, 56, 16;

%e 1, 10, 51, 171, 411, 735, 986, 977, 684, 304, 64;

%e 1, 13, 86, 378, 1219, 3027, 5930, 9254, 11485, 11185, 8304, 4448, 1536, 256;

%p CP:= proc(n, i) option remember;

%p `if`(n=0 and (i=0 or i=1), 1, `if`(n<0 or i<0, 0,

%p CP(n-1, i) +3*CP(n-1, i-1) +5*CP(n-1, i-2) +4*CP(n-1, i-3)))

%p end:

%p seq(seq(CP(n, i), i=0..3*n+1), n=0..6); # _Alois P. Heinz_, Apr 30 2012

%t CP[0, 0] = CP[0, 1] = 1;

%t CP[n_ /; n >= 0, i_] /; 0 <= i <= 3n+1 := CP[n, i] =

%t CP[n-1, i] + 3 CP[n-1, i-1] + 5 CP[n-1, i-2] + 4 CP[n-1, i-3];

%t CP[_, _] = 0;

%t Table[CP[n, i], {n, 0, 6}, {i, 0, 3n+1}] // Flatten (* _Jean-François Alcover_, Feb 14 2021 *)

%Y Cf. A027907.

%K nonn,tabf

%O 0,4

%A Thomas J. Pfaff (tpfaff(AT)ithaca.edu), Oct 02 2006

%E Corrected and extended by _Alois P. Heinz_, Apr 30 2012