OFFSET
1,2
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
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.
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
EXAMPLE
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;
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Sep 05 2008
EXTENSIONS
Typo corrected by Philippe Deléham, Jan 05 2009
STATUS
approved