A182491 T(n,k), a triangular array read by rows, is the Wiener index for the complete bipartite graph K(n,k).

1, 4, 8, 9, 14, 21, 16, 22, 30, 40, 25, 32, 41, 52, 65, 36, 44, 54, 66, 80, 96, 49, 58, 69, 82, 97, 114, 133, 64, 74, 86, 100, 116, 134, 154, 176, 81, 92, 105, 120, 137, 156, 177, 200, 225, 100, 112, 126, 142, 160, 180, 202, 226, 252, 280, 121, 134, 149, 166, 185, 206, 229, 254, 281, 310, 341

Offset

1,2

Comments

T(n,1) = A000290 (star graphs).

T(n,n) = A000567.

Links

Indranil Ghosh, Rows 1..100, flattened

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Eric Weisstein's World of Mathematics, Wiener Index

Formula

T(n,k) = n*k + n*(n-1) + k*(k-1).

G.f.: x*y*(x+y+2*x*y^2+2*x^2*y-7*x*y+1)/((1-x)^3*(1-y)^3). - R. J. Mathar, Nov 27 2015

Example

Triangle of terms:
  1
  4     8
  9     14    21
  16    22    30    40
  25    32    41    52    65
  36    44    54    66    80    96
T(2,1)=4. Indeed, K(2,1) is the graph in the shape of a V and the Wiener index is 2*1 + 2 = 4. - Emeric Deutsch, Aug 25 2013

Maple

Wi := proc (m, n) options operator, arrow: (m+n)^2-m*n-m-n end proc: for m to 10 do seq(Wi(m, n), n = 1 .. m) end do; # yields sequence in triangular form; Emeric Deutsch, Aug 25 2013

Mathematica

Table[Table[3n^2-2*n+(m-n)3n + 2*Binomial[m-n, 2], {n, 1, m}], {m, 1, 20}]//Grid
(* Second program *)
Table[SeriesCoefficient[x y (x + y + 2 x y^2 + 2 x^2*y - 7 x y + 1)/((1 - x)^3*(1 - y)^3), {x, 0, n}, {y, 0, k}], {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, Feb 24 2017 *)

Prog

(Python)
i=1
for n in range(1, 101):
....for k in range(1, n+1):
........print str(i)+" "+str(n*k + n*(n-1) + k*(k-1))
........i+=1 # Indranil Ghosh, Feb 24 2017

Crossrefs

Cf. A000567, A228312.

Sequence in context: A312830 A312831 A085711 * A281686 A122785 A280450

Adjacent sequences: A182488 A182489 A182490 * A182492 A182493 A182494

Keyword

nonn,tabl,easy

Author

Geoffrey Critzer, Patrick Cuddy, May 02 2012

Status

approved