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A123531 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.

%I A123531 #17 Feb 14 2021 13:03:11
%S A123531 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 A123531 304,64,1,13,86,378,1219,3027,5930,9254,11485,11185,8304,4448,1536,
%U A123531 256,1,16,130,705,2835,8918,22618,47055,81005,115630,136300,131225,101140
%N A123531 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 A123531 Alois P. Heinz, <a href="/A123531/b123531.txt">Rows n = 0..81, flattened</a>
%H A123531 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 A123531 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 A123531 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 A123531 Triangle begins:
%e A123531 1,  1;
%e A123531 1,  4,  8,   9,    4;
%e A123531 1,  7, 25,  57,   87,   89,   56,   16;
%e A123531 1, 10, 51, 171,  411,  735,  986,  977,   684,   304,   64;
%e A123531 1, 13, 86, 378, 1219, 3027, 5930, 9254, 11485, 11185, 8304, 4448, 1536, 256;
%p A123531 CP:= proc(n, i) option remember;
%p A123531        `if`(n=0 and (i=0 or i=1), 1, `if`(n<0 or i<0, 0,
%p A123531         CP(n-1, i) +3*CP(n-1, i-1) +5*CP(n-1, i-2) +4*CP(n-1, i-3)))
%p A123531      end:
%p A123531 seq(seq(CP(n, i), i=0..3*n+1), n=0..6); # _Alois P. Heinz_, Apr 30 2012
%t A123531 CP[0, 0] = CP[0, 1] = 1;
%t A123531 CP[n_ /; n >= 0, i_] /; 0 <= i <= 3n+1 := CP[n, i] =
%t A123531      CP[n-1, i] + 3 CP[n-1, i-1] + 5 CP[n-1, i-2] + 4 CP[n-1, i-3];
%t A123531 CP[_, _] = 0;
%t A123531 Table[CP[n, i], {n, 0, 6}, {i, 0, 3n+1}] // Flatten (* _Jean-François Alcover_, Feb 14 2021 *)
%Y A123531 Cf. A027907.
%K A123531 nonn,tabf
%O A123531 0,4
%A A123531 Thomas J. Pfaff (tpfaff(AT)ithaca.edu), Oct 02 2006
%E A123531 Corrected and extended by _Alois P. Heinz_, Apr 30 2012