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.

1, 1, 1, 4, 8, 9, 4, 1, 7, 25, 57, 87, 89, 56, 16, 1, 10, 51, 171, 411, 735, 986, 977, 684, 304, 64, 1, 13, 86, 378, 1219, 3027, 5930, 9254, 11485, 11185, 8304, 4448, 1536, 256, 1, 16, 130, 705, 2835, 8918, 22618, 47055, 81005, 115630, 136300, 131225, 101140

Offset: 0

Author

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

Keywords

• nonn
• tabf

Examples

			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.
Triangle begins:
1,  1;
1,  4,  8,   9,    4;
1,  7, 25,  57,   87,   89,   56,   16;
1, 10, 51, 171,  411,  735,  986,  977,   684,   304,   64;
1, 13, 86, 378, 1219, 3027, 5930, 9254, 11485, 11185, 8304, 4448, 1536, 256;
		

Links

• Alois P. Heinz, Rows n = 0..81, flattened
• T. Pfaff and J. Walker, The Chromatic Polynomial of P_2 X P_n and C_3 X P_n, Missouri J. Math. Sci., 20, Issue 3 (2008), 169-177.

Crossrefs

Cf. A027907.

Programs

Maple

CP:= proc(n, i) option remember;
       `if`(n=0 and (i=0 or i=1), 1, `if`(n<0 or i<0, 0,
        CP(n-1, i) +3*CP(n-1, i-1) +5*CP(n-1, i-2) +4*CP(n-1, i-3)))
     end:
seq(seq(CP(n, i), i=0..3*n+1), n=0..6); # Alois P. Heinz, Apr 30 2012

Mathematica

CP[0, 0] = CP[0, 1] = 1;
CP[n_ /; n >= 0, i_] /; 0 <= i <= 3n+1 := 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];
CP[, ] = 0;
Table[CP[n, i], {n, 0, 6}, {i, 0, 3n+1}] // Flatten (* Jean-François Alcover, Feb 14 2021 *)

Formula

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.

Extensions

Corrected and extended by Alois P. Heinz, Apr 30 2012