A228312 Triangle read by rows: T(m,n) (1<=n<=m) is the hyper-Wiener index of the complete bipartite graph K(m,n).

1, 5, 10, 12, 18, 27, 22, 29, 39, 52, 35, 43, 54, 68, 85, 51, 60, 72, 87, 105, 126, 70, 80, 93, 109, 128, 150, 175, 92, 103, 117, 134, 154, 177, 203, 232, 117, 129, 144, 162, 183, 207, 234, 264, 297, 145, 158, 174, 193, 215, 240, 268, 299, 333, 370, 176, 190, 207, 227, 250, 276, 305

Offset

1,2

Comments

T(n,n) = 4n^2 - 3n = A001107(n).

Links

Table of n, a(n) for n=1..62.

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

Formula

T(m,n) = (3(m+n)^2 - 4mn - 3m - 3n)/2.

The Hosoya-Wiener polynomial of K(m,n) is mnt + (1/2)(m^2 + n^2 - m - n) t^2.

G.f.: x*y*(1+2*x+2*y+3*x^2*y+3*y^2*x-11*x*y)/((1-x)^3*(1-y)^3). - R. J. Mathar, Nov 27 2015

Example

Triangle begins:
    1,
    5, 10,
   12, 18, 27,
   22, 29, 39, 52,
   35, 43, 54, 68, 85,
   51, 60, 72, 87,105,126,
   70, 80, 93,109,128,150,175,
   92,103,117,134,154,177,203,232,
  117,129,144,162,183,207,234,264,297,
  145,158,174,193,215,240,268,299,333,370,

Maple

HWi := proc (m, n) options operator, arrow: (3/2)*(m+n)^2-2*m*n-(3/2)*m-(3/2)*n end proc: for m to 10 do seq(HWi(m, n), n = 1 .. m) end do; # yields sequence in triangular form

Crossrefs

Cf. A001107, A182491.

Sequence in context: A313368 A313369 A101839 * A313370 A313371 A313372

Adjacent sequences: A228309 A228310 A228311 * A228313 A228314 A228315

Keyword

nonn,tabl,easy

Author

Emeric Deutsch, Aug 25 2013

Status

approved