A228308
Triangle read by rows: T(n,k) (n>=2, 1<=k<=n-1) is the number of unordered pairs of vertices at distances k in the odd graph O_n.
Original entry on oeis.org
3, 15, 30, 70, 210, 315, 315, 1260, 2520, 3780, 1386, 6930, 17325, 34650, 46200, 6006, 36036, 108108, 270270, 450450, 600600, 25740, 180180, 630630, 1891890, 3783780, 6306300, 7882875, 109395, 875160, 3500640, 12252240, 28588560, 57177120
Offset: 2
Row 2 has only one entry equal to 3; indeed, O_2 is the complete graph K_3, having 3 distances equal to 1.
- N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161.
- R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531.
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B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: for n from 2 to 10 do seq(coeff(H(n), t, k), k = 1 .. n-1) end do; # yields sequence in triangular form
A228309
The hyper-Wiener index of the odd graph O_n (n>=2).
Original entry on oeis.org
3, 105, 2590, 57015, 1165626, 22834812, 433178460, 8036703675, 146451924190, 2632740298188, 46790614294788, 824017920352900, 14397367664647800, 249906966022292400, 4312825574857068600, 74063143648813911075
Offset: 2
- N. Biggs, Algebraic Graph Theory, Cambridge, 2nd. Ed., 1993, p. 161.
- R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer, The Wiener index of odd graphs, J. Indian. Inst. Sci., vol. 86, 2006, 527-531.
-
B := proc (n) options operator, arrow: [seq(n-floor((1/2)*m), m = 1 .. n-1)] end proc: C := proc (n) options operator, arrow: [seq(ceil((1/2)*m), m = 1 .. n-1)] end proc: H := proc (n) options operator, arrow: (1/2)*binomial(2*n-1, n-1)*(sum((product(B(n)[r]/C(n)[r], r = 1 .. j))*t^j, j = 1 .. n-1)) end proc: HWi := proc (n) options operator, arrow: subs(t = 1, diff(H(n), t)+(1/2)*(diff(H(n), `$`(t, 2)))) end proc: seq(HWi(n), n = 2 .. 20);
A301566
a(n) = Sum_{k=1..n-1} k*A088459(n, k).
Original entry on oeis.org
0, 2, 15, 82, 405, 1891, 8554, 37850, 164985, 710893, 3036726, 12880847, 54331550, 228089538, 953811972, 3975120810, 16519242465, 68474376025, 283211458750, 1169062910873, 4817380232522, 19819870885230, 81429323786460, 334120527783367, 1369374666890230
Offset: 1
-
Table[Sum[k Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {k, n - 1}], {n, 20}]
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T(n, k) = binomial(n, ceil(k/2))*binomial(n-1, k\2);
a(n) = sum(k=1, n-1, k*T(n,k)); \\ Altug Alkan, Mar 23 2018
A228310
The hyper-Wiener index of the hypercube graph Q(n) (n>=2).
Original entry on oeis.org
10, 72, 448, 2560, 13824, 71680, 360448, 1769472, 8519680, 40370176, 188743680, 872415232, 3992977408, 18119393280, 81604378624, 365072220160, 1623497637888, 7181185318912, 31610959298560, 138538465099776, 604731395276800, 2630031813640192
Offset: 2
- Norman Biggs, Algebraic Graph Theory, 2nd ed. Cambridge University Press, 1993 (p. 161).
- R. Balakrishnan, N. Sridharan and K. Viswanathan Iyer,, The Wiener index of odd graphs, J. Ind. Inst. Sci., vol. 86, no. 5, 2006. [Cached copy]
- Eric Weisstein's World of Mathematics, Hypercube Graph.
- Index entries for linear recurrences with constant coefficients, signature (12,-48,64)
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a := proc (n) options operator, arrow: 4^(n-2)*n*(3+n) end proc: seq(a(n), n = 2 .. 25);
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LinearRecurrence[{12,-48,64},{10,72,448},30] (* Harvey P. Dale, Dec 13 2024 *)
Showing 1-4 of 4 results.
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