A245826 Triangle read by rows: T(m,n) is the Szeged index of the grid graph P_m X P_n (1 <= n <= m).

0, 1, 16, 4, 59, 216, 10, 144, 526, 1280, 20, 285, 1040, 2530, 5000, 35, 496, 1809, 4400, 8695, 15120, 56, 791, 2884, 7014, 13860, 24101, 38416, 84, 1184, 4316, 10496, 20740, 36064, 57484, 86016, 120, 1689, 6156, 14970, 29580, 51435, 81984, 122676, 174960, 165, 2320, 8455, 20560, 40625, 70640, 112595, 168480, 240285, 330000

Offset: 1

Author

Emeric Deutsch, Aug 06 2014

Keywords

• nonn
• tabl

Comments

T(n,1) = Szeged index of the path tree P_n = A000292(n-1).

T(n,2) = Szeged index of the ladder graph P_2 X P_n = A063521(n).

T(n,3) = Szeged index of the grid graph P_3 X P_n = A245827(n).

T(n,n) = Szeged index of the grid graph P_n X P_n = A245828(n).

Examples

			T(2,2) = 16 because P_2 X P_2 is the square C_4 and each of its 4 edges contributes 2*2=4 to its Szeged index.
Triangle starts:
0;
1,16;
4,59,216;
10,144,526,1280;
20,285,1040,2530,5000;
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett., 9, 1996, 45-49.

Crossrefs

Cf. A000292, A063521, A245827, A245828.

Cf. A245940 (row sums), A245941 (central terms).

Programs

Haskell

a245826 n k = n * k * (2 * n^2 * k^2 - n^2 - k^2) `div` 6
a245826_row n = map (a245826 n) [1..n]
a245826_tabl = map a245826_row [1..]
-- Reinhard Zumkeller, Aug 07 2014

Maple

T:=proc(m,n) options operator, arrow: (1/6)*m*n*(2*m^2*n^2-m^2-n^2) end proc: for m to 12 do seq(T(m, n), n = 1 .. m) end do; # yields sequence in triangular form

Mathematica

T[m_, n_] := (1/6)*m*n*(2*m^2*n^2 - m^2 - n^2); Table[T[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Feb 09 2018 *)

Formula

T(m,n) = mn(2m^2 n^2 - m^2 - n^2)/6. See the Klavzar et al. reference; p. 47, line 6; there is a typo: n^2 - m^2 should be n^2 + m^2.