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Szeged index of the grid graph P_n X P_n.
+10
4
0, 16, 216, 1280, 5000, 15120, 38416, 86016, 174960, 330000, 585640, 988416, 1599416, 2497040, 3780000, 5570560, 8018016, 11302416, 15638520, 21280000, 28523880, 37715216, 49252016, 63590400, 81250000, 102819600, 128963016, 160425216, 198038680, 242730000, 295526720, 357564416, 430094016, 514489360
FORMULA
a(n) = (1/3)*n^4*(n^2 - 1).
G.f.: 8*x^2*(1+x)*(2+11*x+2*x^2)/(1-x)^7.
Sum_{n>=2} 1/a(n) = 33/4 - Pi^2/2 - Pi^4/30.
Sum_{n>=2} (-1)^n/a(n) = 7*Pi^4/240 + Pi^2/4 - 21/4. (End)
MAPLE
a := proc (n) options operator, arrow: (1/3)*n^4*(n^2-1) end proc: seq(a(n), n = 1 .. 40);
MATHEMATICA
CoefficientList[Series[8 x (1 + x) (2 + 11 x + 2 x^2)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *)
Triangle read by rows: T(n,k) is the Wiener index of a k X n grid (i.e., P_k X P_n, where P_m is the path graph on m vertices; 1 <= k <= n).
+10
3
0, 1, 8, 4, 25, 72, 10, 56, 154, 320, 20, 105, 280, 570, 1000, 35, 176, 459, 920, 1595, 2520, 56, 273, 700, 1386, 2380, 3731, 5488, 84, 400, 1012, 1984, 3380, 5264, 7700, 10752, 120, 561, 1404, 2730, 4620, 7155, 10416, 14484, 19440
COMMENTS
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph.
This is the lower triangular half of a symmetric square array.
FORMULA
T(n,k) = k*n*(n+k)*(k*n-1)/6 (k, n >= 1).
EXAMPLE
Presentation as symmetric square array starts:
======================================================
n\k| 1 2 3 4 5 6 7 8 9
---|--------------------------------------------------
1 | 0 1 4 10 20 35 56 84 120 ...
2 | 1 8 25 56 105 176 273 400 561 ...
3 | 4 25 72 154 280 459 700 1012 1404 ...
4 | 10 56 154 320 570 920 1386 1984 2730 ...
5 | 20 105 280 570 1000 1595 2380 3380 4620 ...
6 | 35 176 459 920 1595 2520 3731 5264 7155 ...
7 | 56 273 700 1386 2380 3731 5488 7700 10416 ...
8 | 84 400 1012 1984 3380 5264 7700 10752 14484 ...
9 | 120 561 1404 2730 4620 7155 10416 14484 19440 ...
T(2,2)=8 because in a square we have four distances equal to 1 and two distances equal to 2.
T(2,1)=1 because on the path graph on two vertices there is one distance equal to 1.
T(3,2)=25 because on the P(2) X P(3) graph there are 7 distances equal to 1, 6 distances equal to 2 and 2 distances equal to 3, with 7*1 + 6*2 + 2*3 = 25.
Triangle starts: 0; 1,8; 4,25,72; 10,56,154,320;
MAPLE
T:=proc(n, k) options operator, arrow: (1/6)*k*n*(n+k)*(k*n-1) end proc: for n to 9 do seq(T(n, k), k=1..n) end do; # yields sequence in triangular form
MATHEMATICA
Table[k n (n + k) (k n - 1)/6, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, May 28 2017 *)
PROG
(PARI)
T(n, k)=k*n*(n+k)*(k*n-1)/6;
for (n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, May 27 2017
Szeged index of the grid graph P_3 X P_n.
+10
3
4, 59, 216, 526, 1040, 1809, 2884, 4316, 6156, 8455, 11264, 14634, 18616, 23261, 28620, 34744, 41684, 49491, 58216, 67910, 78624, 90409, 103316, 117396, 132700, 149279, 167184, 186466, 207176, 229365, 253084, 278384, 305316, 333931, 364280, 396414, 430384, 466241, 504036, 543820
FORMULA
a(n) = (1/2)*n*(17*n^2 - 9).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: x*(4*x^2+43*x+4) / (x-1)^4. - Colin Barker, Aug 07 2014
MAPLE
a := proc (n) options operator, arrow: (1/2)*n*(17*n^2-9) end proc: seq(a(n), n = 1 .. 40);
MATHEMATICA
CoefficientList[Series[(4 x^2 + 43 x + 4)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *) LinearRecurrence[{4, -6, 4, -1}, {4, 59, 216, 526}, 40] (* Harvey P. Dale, Oct 21 2017 *)
PROG
(PARI) Vec(x*(4*x^2+43*x+4)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 07 2014
(2n^7 + 4n^6 - n^5 - 4n^4 - n^3) / 24.
+10
2
0, 0, 17, 279, 1960, 8875, 30555, 87122, 216384, 483570, 994125, 1909985, 3469752, 6013189, 10010455, 16096500, 25111040, 38144532, 56590569, 82205115, 117173000, 164182095, 226505587, 308092774, 413668800, 548843750, 720231525, 935578917, 1203905304
COMMENTS
For n > 0: sum of n-th row of triangle A245826.
FORMULA
a(n) = n^3*(2*n^3 + 2*n^2 - 3*n - 1)*(n + 1)/24 = n^3*(n - 1)*(n + 1)*(2*n^2 + 4*n + 1)/24.
G.f.: x^2*(x^4 + 55*x^3 + 204*x^2 + 143*x + 17) / (x - 1)^8. - Colin Barker, Aug 08 2014
MATHEMATICA
Table[(2 n^7 + 4 n^6 - n^5 - 4 n^4 - n^3)/24, {n, 0, 30}] (* Vincenzo Librandi, Aug 09 2014 *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 17, 279, 1960, 8875, 30555, 87122}, 30] (* Harvey P. Dale, Apr 19 2018 *)
PROG
(Haskell)
a245940 n = n^3 * (2 * n^3 + 2 * n^2 - 3 * n - 1) * (n + 1) `div` 24
(PARI)
concat([0, 0], Vec(x^2*(x^4+55*x^3+204*x^2+143*x+17)/(x-1)^8 + O(x^100))) \\ Colin Barker, Aug 08 2014 (Magma) [(2*n^7 + 4*n^6 - n^5 - 4*n^4 - n^3) / 24: n in [0..30]] // Vincenzo Librandi, Aug 09 2014
(16n^6 - 24n^5 + 2n^4 + 11n^3 - 6n^2 + n) / 6.
+10
2
0, 0, 59, 1040, 7014, 29580, 94105, 247884, 570220, 1184424, 2271735, 4087160, 6977234, 11399700, 17945109, 27360340, 40574040, 58723984, 83186355, 115606944, 157934270, 212454620, 281829009, 369132060, 477892804, 612137400, 776433775, 975938184, 1216443690
COMMENTS
For n > 0: a(n) = A245826(2*n-1,n), central terms of triangle A245826.
FORMULA
G.f.: -x^2*(4*x^4+257*x^3+973*x^2+627*x+59) / (x-1)^7. - Colin Barker, Aug 08 2014 a(0)=0, a(1)=0, a(2)=59, a(3)=1040, a(4)=7014, a(5)=29580, a(6)=94105, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Apr 13 2015
MATHEMATICA
Table[(16 n^6 - 24 n^5 + 2 n^4 + 11 n^3 - 6 n^2 + n)/6, {n, 0, 30}] (* Vincenzo Librandi, Aug 09 2014 *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 59, 1040, 7014, 29580, 94105}, 30] (* Harvey P. Dale, Apr 13 2015 *)
PROG
(Haskell)
a245941 n = n * (16*n^5 - 24*n^4 + 2*n^3 + 11*n^2 - 6*n + 1) `div` 6
(PARI)
concat([0, 0], Vec(-x^2*(4*x^4+257*x^3+973*x^2+627*x+59)/(x-1)^7 + O(x^100))) \\ Colin Barker, Aug 08 2014 (Magma) [(16*n^6-24*n^5+2*n^4+11*n^3-6*n^2+n)/6: n in [0..30]] // Vincenzo Librandi, Aug 09 2014
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