A085537 -id:A085537 - cp's OEIS frontend

A270228 Number of matchings in the n X n rook graph K_n X K_n.

1, 7, 370, 270529, 3337807996, 855404716021831, 5352265402523357926168, 940288991338542314571521981185, 5236753179470435264288904589157765055760, 1029720447530443779943631183186535523331685533812231

Offset: 1

Andrew Howroyd, Mar 13 2016

nonn

K_n X K_n is also called the rook graph or lattice graph.

Andrew Howroyd, Table of n, a(n) for n = 1..16
Andrew Howroyd, C# software related to this sequence
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Rook Graph
Wikipedia, Rook's graph, Hosoya index

Cf. A270227, A270229, A085537 (Wiener index), A002720 (independent vertex sets), A269561, A028420.

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A163284 Triangle read by rows in which row n lists n+1 terms, starting with n^4 and ending with n^5, such that the difference between successive terms is equal to n^4 - n^3.

Original entry on oeis.org

0, 1, 1, 16, 24, 32, 81, 135, 189, 243, 256, 448, 640, 832, 1024, 625, 1125, 1625, 2125, 2625, 3125, 1296, 2376, 3456, 4536, 5616, 6696, 7776, 2401, 4459, 6517, 8575, 10633, 12691, 14749, 16807, 4096, 7680, 11264, 14848, 18432, 22016, 25600, 29184, 32768

Offset: 0

Omar E. Pol, Jul 24 2009

The first term of row n is A000583(n) and the last term of row n is A000584(n).

			Triangle begins:
0;
1,1;
16,24,32;
81,135,189,243;
256,448,640,832,1024;
625,1125,1625,2125,2625,3125;
1296,2376,3456,4536,5616,6696,7776;
2401,4459,6517,8575,10633,12691,14749,16807;
4096,7680,11264,14848,18432,22016,25600,29184,32768;
6561,12393,18225,24057,29889,35721,41553,47385,53217,59049;
10000,19000,28000,37000,46000,55000,64000,73000,82000,91000,100000;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000583, A000584, A085537, A159797, A162611, A162614, A162622, A163282, A163283, A163285.

Table[n^4 + k*(n^4 - n^3), {n,0,15}, {k,0,n}] // Flatten (* G. C. Greubel, Dec 17 2016 *)

A163284(n, k)=n^4 +k*(n^4 -n^3) \\ G. C. Greubel, Dec 17 2016

A240930 a(n) = n^7 - n^6.

Original entry on oeis.org

0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008, 4251528, 9000000, 17715610, 32845824, 57921708, 97883968, 159468750, 251658240, 386201104, 578207808, 846825858, 1216000000, 1715322420, 2380977984, 3256789558, 4395368448, 5859375000, 7722894400, 10072932714

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 7-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001014, A001015.

Cf. A002378, A045991, A085537, A085538, A085539, A240931, A240932, A240933.

[n^7-n^6 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240930:=n->n^7-n^6: seq(A240930(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^7 - n^6, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)

vector(100, n, (n-1)^7 - (n-1)^6) \\ Derek Orr, Aug 03 2014

a(n) = n^6*(n-1) = n^7 - n^6.
a(n) = A001015(n) - A001014(n).
G.f.: 2*(32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 6 - Sum_{k=2..6} zeta(k). - Amiram Eldar, Jul 05 2020

A240931 a(n) = n^8 - n^7.

Original entry on oeis.org

0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064, 38263752, 90000000, 194871710, 394149888, 752982204, 1370375552, 2392031250, 4026531840, 6565418768, 10407740544, 16089691302, 24320000000, 36021770820, 52381515648, 74906159834, 105488842752, 146484375000

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 8-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A001015, A001016.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240932, A240933.

[n^8-n^7 : n in [0..30]]; // Wesley Ivan Hurt, Aug 09 2014

A240931:=n->n^8-n^7: seq(A240931(n), n=0..30); # Wesley Ivan Hurt, Aug 09 2014

Table[n^8 - n^7, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 09 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,128,4374,49152,312500,1399680,4941258,14680064},30] (* Harvey P. Dale, Apr 29 2016 *)

vector(100, n, (n-1)^8 - (n-1)^7) \\ Derek Orr, Aug 03 2014

concat([0,0], Vec(-2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9 + O(x^100))) \\ Colin Barker, Aug 08 2014

a(n) = n^7*(n-1) = n^8 - n^7.
a(n) = A001016(n) - A001015(n).
G.f.: -2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 7 - Sum_{k=2..7} zeta(k). - Amiram Eldar, Jul 05 2020

A240932 a(n) = n^9 - n^8.

Original entry on oeis.org

0, 0, 256, 13122, 196608, 1562500, 8398080, 34588806, 117440512, 344373768, 900000000, 2143588810, 4729798656, 9788768652, 19185257728, 35880468750, 64424509440, 111612119056, 187339329792, 305704134738, 486400000000, 756457187220, 1152393344256, 1722841676182

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 9-digit positive integers in base n.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A001016, A001017.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240933.

[n^9-n^8: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

Table[n^9 - n^8, {n, 0, 40}] (* Vincenzo Librandi, Aug 15 2016 *)

vector(100, n, (n-1)^9 - (n-1)^8) \\ Derek Orr, Aug 03 2014

a(n) = n^8*(n-1) = n^9 - n^8.
a(n) = A001017(n) - A001016(n).
G.f.: 2*x^2*(x^7+374*x^6+9327*x^5+49780*x^4+78095*x^3+38454*x^2+5281*x+128) / (x-1)^10. - Colin Barker, Aug 08 2014
Sum_{n>=2} 1/a(n) = 8 - Sum_{k=2..8} zeta(k). - Amiram Eldar, Jul 05 2020

A240933 a(n) = n^10 - n^9.

Original entry on oeis.org

0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632

Offset: 0

Martin Renner, Aug 03 2014

For n>1 number of 10-digit positive integers in base n.

Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A001017, A008454.

Cf. A002378, A045991, A085537, A085538, A085539, A240930, A240931, A240932.

[n^10-n^9 : n in [0..30]]; // Wesley Ivan Hurt, Aug 03 2014

A240933:=n->n^10-n^9: seq(A240933(n), n=0..30); # Wesley Ivan Hurt, Aug 03 2014

Table[n^10 - n^9, {n, 0, 30}] (* Wesley Ivan Hurt, Aug 03 2014 *)
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,512,39366,786432,7812500,50388480,242121642,939524096,3099363912,9000000000},40] (* Harvey P. Dale, Oct 19 2022 *)

vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014

a(n) = n^9*(n-1) = n^10 - n^9.
a(n) = A008454(n) - A001017(n). - Michel Marcus, Aug 03 2014
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014
Recurrence: a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Aug 03 2014
Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020

A085540 a(n) = n*(n + 1)^3.

Original entry on oeis.org

0, 8, 54, 192, 500, 1080, 2058, 3584, 5832, 9000, 13310, 19008, 26364, 35672, 47250, 61440, 78608, 99144, 123462, 152000, 185220, 223608, 267674, 317952, 375000, 439400, 511758, 592704, 682892, 783000, 893730, 1015808, 1149984, 1297032, 1457750, 1632960

Offset: 0

N. J. A. Sloane, Jul 05 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A085537 (same sequence with a 0 prepended), A092364.

[n*(n+1)^3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2016

Table[n (n + 1)^3, {n, 0, 40}] (* Vincenzo Librandi, Aug 15 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{0,8,54,192,500},40] (* Harvey P. Dale, May 06 2019 *)

a(n) = 2*A092364(n+1). - Zerinvary Lajos, May 09 2007
G.f.: -2*x*(4 + 7*x + x^2)/(x - 1)^5. - R. J. Mathar, Mar 10 2011
a(n) = A085537(n-1). - Eric W. Weisstein, Sep 08 2017
E.g.f.: exp(x)*x*(8 + 19*x + 9*x^2 + x^3). - Stefano Spezia, Jun 10 2023
From Amiram Eldar, Jul 02 2023: (Start)
Sum_{n>=1} 1/a(n) = 3 - Pi^2/6 - zeta(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 + 2*log(2) + 3*zeta(3)/4 - 3. (End)

A330520 Sum of even integers <= n times the sum of odd integers <= n.

Original entry on oeis.org

0, 0, 2, 8, 24, 54, 108, 192, 320, 500, 750, 1080, 1512, 2058, 2744, 3584, 4608, 5832, 7290, 9000, 11000, 13310, 15972, 19008, 22464, 26364, 30758, 35672, 41160, 47250, 54000, 61440, 69632, 78608, 88434, 99144, 110808, 123462, 137180, 152000, 168000, 185220, 203742, 223608, 244904, 267674

Offset: 0

J. Stauduhar, Dec 17 2019

Number of crossings in a grid formed by drawing n parallel infinite-length lines perpendicular to the previous number of lines.

The sum of odd integers <= n is m^2 where m = round(n/2) is their number. The sum of even integers <= n is k(k+1) where k = floor(n/2) is their number. So a(n) = m^2*k(k+1), where the factor m appears three times. - M. F. Hasler, Dec 19 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A179824, A085537, A007009, A002620.

Cf. A000290 (sum of odd integers), A002378 (sum of even integers).

CoefficientList[Series[2 (x^2 + x + 1) x^2/((x + 1)^2*(1 - x)^5), {x, 0, 45}], x] (* Michael De Vlieger, Dec 22 2019 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,2,8,24,54,108},50] (* Harvey P. Dale, Dec 29 2021 *)

apply( A330520(n)=n\2*(n\2+1)*(n\/2)^2, [0..99]) \\ M. F. Hasler, Dec 19 2019

G.f.: 2*(x^2+x+1)*x^2/((x+1)^2*(1-x)^5).
a(n) = 2 * A007009(n-1) for n>1.
a(2k+i) = (k+i)^3 (k+1-i), with i = 0 or 1. - M. F. Hasler, Dec 19 2019
a(n) = A002378(floor(n/2)) * A000290(ceiling(n/2)). - Bernard Schott, Dec 22 2019

A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

Original entry on oeis.org

0, 1, 7, 2, 15, 26, 3, 23, 53, 63, 4, 31, 80, 127, 124, 5, 39, 107, 191, 249, 215, 6, 47, 134, 255, 374, 431, 342, 7, 55, 161, 319, 499, 647, 685, 511, 8, 63, 188, 383, 624, 863, 1028, 1023, 728, 9, 71, 215, 447, 749, 1079, 1371, 1535, 1457, 999, 10, 79, 242, 511, 874, 1295, 1714, 2047, 2186, 1999, 1330

Offset: 2

Stefano Spezia, Jul 11 2020

T(n, k) is a sharp upper bound of the tree width of a graph G that does not contain a clique on n vertices nor a minimal separator of size larger than k (see Theorem 2.1 in Pilipczuk et al.).

All the square matrices starting at top left of the table T are singular except for the 2 X 2 submatrix: det([0, 7; 1, 15]) = -7.

			The table starts at row n = 2 and column k = 1 as:
0   7   26   63  124   215 ...
1  15   53  127  249   431 ...
2  23   80  191  374   647 ...
3  31  107  255  499   863 ...
4  39  134  319  624  1079 ...
5  47  161  383  749  1295 ...
...

Marcin Pilipczuk, Ni Luh Dewi Sintiari, Stéphan Thomassé and Nicolas Trotignon, (Theta, triangle)-free and (even hole, K4)-free graphs. Part 2 : bounds on treewidth, arXiv:2001.01607 [cs.DM], 2020. See p. 7.
Index entries for sequences related to trees.

Cf. A000578, A001093, A001477 (k = 1), A004771 (k = 2), A068601 (n = 2), A085537, A109129, A123865 (main diagonal), A325543, A325612.

T[n_,k_]:=(n-1)*k^3-1; Flatten[Table[T[n+1-k,k],{n,2,12},{k,1,n-1}]]

PARI
```
T(n, k) = (n - 1)*k^3 - 1
```

O.g.f.: x^2*y*(y*(7 - 2*y + y^2) + x*(1 - y)^3)/((1 - x)^2*(1 - y)^4).
E.g.f.: -1 + exp(x) - x + exp(y)*x + exp(y)*(1 + y + 3*y^2 + y^3) + exp(x + y)*(-1 +(-1 + x)*y*(1 + 3*y + y^2)).
T(n, k) = n*A000578(k) - A001093(k).
T(n, n) = A085537(n) - 1 for n > 1.
T(n, k) = T(n+1, 1)*T(2, k) + T(n, 1).

cp's OEIS Frontend

A270228 Number of matchings in the n X n rook graph K_n X K_n.

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A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A163284 Triangle read by rows in which row n lists n+1 terms, starting with n^4 and ending with n^5, such that the difference between successive terms is equal to n^4 - n^3.

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A240930 a(n) = n^7 - n^6.

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A240931 a(n) = n^8 - n^7.

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A240932 a(n) = n^9 - n^8.

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A240933 a(n) = n^10 - n^9.

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