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 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). 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Cf. A002697, A006516. Sequence in context: A152664 A167591 A227043 * A111667 A323825 A019239 Adjacent sequences:  A143373 A143374 A143375 * A143377 A143378 A143379 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Sep 05 2008 EXTENSIONS Typo corrected by Philippe Deléham, Jan 05 2009 STATUS approved

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Last modified January 17 12:33 EST 2020. Contains 330958 sequences. (Running on oeis4.)