A143376 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1 <= k <= n).

1, 4, 2, 12, 12, 4, 32, 48, 32, 8, 80, 160, 160, 80, 16, 192, 480, 640, 480, 192, 32, 448, 1344, 2240, 2240, 1344, 448, 64, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256, 5120, 23040, 61440, 107520

Offset: 1

Author

Emeric Deutsch, Sep 05 2008

Keywords

• nonn
• tabl

Comments

Sum of entries in row n = 2^(n-1)*(2^n-1) = A006516.

The entries in row n are the coefficients of the Wiener polynomial of the cube Q_n.

Sum_{k=1..n} k*T(n,k) = n*4^(n-1) = A002697(n) = the Wiener index of the cube Q_n.

Triangle T(n,k), 1 <= k <= n, read by rows given by [1,1,0,0,0,0,0,...]DELTA[1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938; subtriangle of triangle A055372. - Philippe Deléham, Oct 14 2008

Examples

			T(2,1)=4, T(2,2)=2 because in Q_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2.
Triangle starts:
   1;
   4,   2;
  12,  12,   4;
  32,  48,  32,   8;
  80, 160, 160,  80,  16;
		

Links

• Indranil Ghosh, Rows 1..125, 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.

Crossrefs

Cf. A002697, A006516.

Programs

Maple

T:=proc(n,k) options operator, arrow: 2^(n-1)*binomial(n,k) end proc: for n to 10 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Mathematica

nn = 8; A[u_, z_] := (z + u z)/(1 - (z + u z));
Drop[Map[Select[#, # > 0 &] &, Map[Drop[#, 1] &,CoefficientList[Series[1/(1 - A[u, z]), {z, 0, nn}], {z, u}]]],1] // Grid (* Geoffrey Critzer, Mar 04 2017 *)
Flatten[Table[2^(n-1) Binomial[n, k], {n, 10},{k,n}]] (* Indranil Ghosh, Mar 06 2017 *)

PARI

tabl(nn) = {for (n=1, nn, for(k=1, n, print1(2^(n-1) * binomial(n, k),", ");); print();); };
tabl(10); \\ Indranil Ghosh, Mar 06 2017

Python

import math
f=math.factorial
def C(n,r): return f(n) / f(r) / f(n-r)
i=1
for n in range(1,126):
    for k in range(1,n+1):
        print(str(i)+" "+str(2**(n-1)*C(n,k)))
        i+=1 # Indranil Ghosh, Mar 06 2017

Formula

T(n,k) = 2^(n-1)*binomial(n,k).

G.f.: G(q,z) = qz/((1-2z)(1-2z-2zq)).

T(n,k) = A055372(n,k). - Philippe Deléham, Oct 14 2008

Extensions

Typo corrected by Philippe Deléham, Jan 05 2009