A019582 -id:A019582 - cp's OEIS frontend

A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 12, 18, 16, 5, 1, 12, 54, 40, 25, 6, 1, 40, 72, 160, 75, 36, 7, 1, 28, 405, 280, 375, 126, 49, 8, 1, 96, 390, 2176, 825, 756, 196, 64, 9, 1, 104, 1944, 2800, 8125, 2016, 1372, 288, 81, 10, 1, 280, 3411, 17920, 13175, 23976, 4312, 2304, 405

Offset: 1

N. J. A. Sloane, Dec 01 2007, based on email from Max Alekseyev, Nov 08 2007

From Andrew Howroyd, Apr 22 2017: (Start)
Number of step shifted (decimated) sequences of length n using a maximum of m different symbols. See A056371 for an explanation of step shifts. -
Number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence.  Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {0..n-1}. (End)

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, ...
3, 9, 18, 54, 72, 405, 390, 1944, 3411, 14985, 17802, 139968, 133104, 798525, 1804518, 5454378, 8072532, 64599849, 64573626, 437732424, ...
4, 16, 40, 160, 280, 2176, 2800, 17920, 44224, 263296, 419872, 4280320, 5594000, 44751616, 134391040, 539054080, 1073758360, 11453771776, 15271054960, 137575813120, ...
5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1830
Max Alekseyev, First 20 rows and columns of table
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row m=2 is A056371
Row m=3 is A056372
Row m=4 is A056373
Row m=5 is A056374
Row m=6 is A056375
Column n=2 is A000290
Column n=3 is A002411
Column n=4 is A019582
Cf. A285522, A285548.

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n]==1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; Table[a[m-n+1, n], {m, 1, 15}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Dec 01 2015 *)

for(i=1,15,for(m=1,i,n=i-m+1; print1(sum(k=1, n, if(gcd(k,n)==1, m^sumdiv(n,d,eulerphi(d)/znorder(Mod(k,d))),0))/eulerphi(n)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

Offset corrected by Andrew Howroyd, Apr 20 2017

A214011 T(n,k) is the number of n X n nonnegative integer matrices with row and column i=1..n having sum <= i*k.

Original entry on oeis.org

2, 3, 12, 4, 54, 271, 5, 160, 7722, 24950, 6, 375, 85639, 9318805, 9800058, 7, 756, 564041, 641631566, 98721672541, 16942485560, 8, 1372, 2663506, 17609835599, 69768979161580, 9463992096711104, 131898088386405, 9, 2304, 9976732

Offset: 1

R. H. Hardin, Jun 30 2012

Table starts
        2,           3,              4,                 5,                  6;
       12,          54,            160,               375,                756;
      271,        7722,          85639,            564041,            2663506;
    24950,     9318805,      641631566,       17609835599,       269462676001;
  9800058, 98721672541, 69768979161580, 11798463876314995, 807203255071567008.
From Robert Israel, Jul 01 2020: (Start)
T(n,k) is the number of integer lattice points in kP where P is an (n^2)-dimensional polytope with vertices having integer coordinates.  Therefore row n is an Ehrhart polynomial in k, with degree n^2 and rational coefficients. (End)

			Some solutions for n=3, k=1:
   0 0 1   0 0 0   0 0 0   0 1 0   0 0 0   1 0 0   1 0 0
   1 0 0   0 1 1   0 0 1   0 0 1   0 1 1   0 1 1   0 0 2
   0 0 1   0 0 1   1 2 0   0 0 0   0 1 1   0 0 2   0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..41

Row 2 is A019582(n+2). Rows 3 to 5: A214012, A214013, A214014.

Empirical: rows 1 2 3 are polynomials of degree 1 4 9.

A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

Original entry on oeis.org

0, 1, 16793600, 423651479175, 562950490292224, 149011627197265625, 14215144250057342976, 670534312205763205375, 18889465949070766899200, 358948993948871860432449, 5000000000500000000000000, 54173529719030485105622951, 476981083228048575587942400

Offset: 0

David Nacin, Feb 27 2017

Cycle index of symmetric group S2 acting on the set of 25 entries is (s(2)^10*s(1)^5 + s(1)^25)/2.

			For n=2 we get a(2)=16793600 inequivalent 5x5 binary matrices up to the action of transposition.

Cf. A282612,A282613,A282614. A283026, A283027, A283028, A283030, A283031, A283032, A283033. A170798 (4x4 version). A168555 (3x3 version). A019582 (2x2 version)

Table[n^15 (n^2 + 1) (n^8 - n^6 + n^4 - n^2 + 1)/2, {n, 0, 12}]

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2; \\ Indranil Ghosh, Feb 27 2017

def A283029(n): return n**15*(n**2+1)*(n**8-n**6+n**4-n**2+1)/2 # Indranil Ghosh, Feb 27 2017

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 16793574*x^23 + 423214845900*x^22 + 551941009751074*x^21 + 134512557517054626*x^20 + 10522699609491808746*x^19 + 347912001753554722204*x^18 + 5696453728178627889150*x^17 + 50977946159336791604079*x^16 + 265857130683340877431996*x^15 + 842694350441988138095256*x^14 + 1667306282568523129263444*x^13 + 2089823554970188253479900*x^12 + 1667306282568523129263444*x^11 + 842694350441988138095256*x^10 + 265857130683340877431996*x^9 + 50977946159336791604079*x^8 + 5696453728178627889150*x^7 + 347912001753554722204*x^6 + 10522699609491808746*x^5 + 134512557517054626*x^4 + 551941009751074*x^3 + 423214845900*x^2 + 16793574*x + 1)/(x - 1)^26. (End)

A179824 Chromatic polynomial of the star graph on 4 vertices (claw graph) and the path graph on 4 vertices.

Original entry on oeis.org

2, 24, 108, 320, 750, 1512, 2744, 4608, 7290, 11000, 15972, 22464, 30758, 41160, 54000, 69632, 88434, 110808, 137180, 168000, 203742, 244904, 292008, 345600, 406250, 474552, 551124, 636608, 731670, 837000, 953312, 1081344, 1221858

Offset: 2

Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 28 2010

To generate a Pythagorean triangle one uses (a,b) to get sides b^2-a^2, 2*a*b, and a^2+b^2, having a perimeter of 2*b*(a+b). If for a one uses the triangular number n*(n+1)/2 and for b the next triangular number (n+1)*(n+2)/2, the perimeter of the triangle so formed is (n+1)^3 * (n+2), which will give the same results as this sequence starting at the second term. - J. M. Bergot, Apr 01 2012
Define b(0)=0 and b(n)=A179824(n+1) for n > 0. Then b(n) is the number of 4-tuples (w,x,y,z) having all terms in {0,...,n} and no two consecutive terms equal. - Clark Kimberling, May 31 2012
Let n points in the plane each become the centers of n-1 concentric circles, circles that pass through only one of each of the other points. The maximum number of intersections of these circles is this sequence. [The solution was given by Andrew Weimholt in the Sequence Fans Mailing List] - J. M. Bergot, Mar 10 2014
Both the 'claw graph', a graph with 4 vertices where one vertex is adjacent to the other three, and the path graph on 4 vertices (per Clark Kimberling's comment), have this sequence as their chromatic polynomial, or the number of proper colorings of the graph using at most n colors. This is the standard example of two graphs which are not isomorphic, but which have the same chromatic polynomial.

			From _Jack W Grahl_, Jul 16 2018: (Start)
Consider the claw graph, which has vertices A, B, C, D, and edges AB, AC, AD. To color this graph with 3 colors, we can choose any of the 3 colors for A. Then each of the other vertices can be colored with any of the remaining two colors, giving 3 * 2 * 2 * 2 = 24 choices in all.
Similarly, consider the path graph with the same vertices and edges AB, BC, CD. We have 3 choices for the color of A, then 2 choices for the color of B (any color except that chosen for A), 2 choices for the color of C (any color except B's) etc. (End)

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Andrew Weimholt, Re: Intersecting circles, SeqFan post Mar 09 2014.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019582, A179824.

List([2..40], n-> n*(n-1)^3 ); # G. C. Greubel, Aug 10 2019

let f n = [ (x,a,b,c) | let t = [1..n], x <- t, a <- t, x /= a, b <- t, x /= b, c <- t, x /= c ] in map (length.f) [2..]

Haskell
```
let f n = n*(n-1)^3 in map f [2..]
```

I:=[2,24,108,320,750]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Mar 12 2014

CoefficientList[Series[2 (1 + 7 x + 4 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)
Table[n^3+n^4,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{2,24, 108,320,750},40] (* Harvey P. Dale, Sep 05 2015 *)

a(n) = n*(n-1)^3 \\ Charles R Greathouse IV, Mar 11 2014

[n*(n-1)^3 for n in (2..40)] # G. C. Greubel, Aug 10 2019

a(n) = n*(n-1)^3. - Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 30 2010
G.f.: 2*x^2*(1 + 7*x + 4*x^2)/(1-x)^5. - Colin Barker, Jan 30 2012
a(n) = 2*A019582(n). - R. J. Mathar, Jun 09 2013
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Mar 12 2014
Sum_{n>=2} 1/a(n) = A249649. - R. J. Mathar, Oct 18 2019
Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/4 + 2*log(2) - Pi^2/12 - 1. - Amiram Eldar, Nov 05 2020

Name edited by Jack W Grahl, Jul 16 2018

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

A364922 a(n) is the square of the side length of a simplex whose n-dimensional inner hypervolume is equal to its (n-1)-dimensional surface hypervolume. As a result, the sequence starts at n=2.

Original entry on oeis.org

48, 216, 640, 1500, 3024, 5488, 9216, 14580, 22000, 31944, 44928, 61516, 82320, 108000, 139264, 176868, 221616, 274360, 336000, 407484, 489808, 584016, 691200, 812500, 949104, 1102248, 1273216, 1463340, 1674000, 1906624, 2162688, 2443716, 2751280, 3087000

Offset: 2

Matt Moir, Apr 13 2024

Setting the generalized hypervolume formula equal to the surface hypervolume formula and solving for the side length x (and ignoring the x = 0 solution, as it would correspond to a simplex consisting of only a single point) gives x = sqrt(2*(n^3)*(n+1)).

Harvey P. Dale, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A019582, A179824.

Table[2*n^3*(n + 1), {n, 2, 50}] (* Paolo Xausa, Apr 18 2024 *)
LinearRecurrence[{5,-10,10,-5,1},{48,216,640,1500,3024},40] (* Harvey P. Dale, Aug 27 2024 *)

def a(n): return 2 * n**3 * (n + 1)
print([a(n) for n in range(2, 50)])

a(n) = 2*n^3*(n+1) = 2*A179824(n+1).
From Stefano Spezia, Apr 13 2024: (Start)
G.f.: 4*x^2*(12 - 6*x + 10*x^2 - 5*x^3 + x^4)/(1 - x)^5.
a(n) = 4*A019582(n+1). (End)

A242983 n/2 * (n^3 - 2n^2 - 2n + 5).

Original entry on oeis.org

0, 1, 1, 12, 58, 175, 411, 826, 1492, 2493, 3925, 5896, 8526, 11947, 16303, 21750, 28456, 36601, 46377, 57988, 71650, 87591, 106051, 127282, 151548, 179125, 210301, 245376, 284662, 328483, 377175, 431086, 490576, 556017

Offset: 0

Ralf Stephan, Jun 09 2014

For n>1, number of ways to place two dominoes horizontally on an n X n chessboard.

Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A242861.

Table[n/2 (n^3-2n^2-2n+5),{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,1,12,58},40] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = A019582(n) + A077414(n-2), n>1.

G.f.: x*(-2*x^3 + 17*x^2 - 4*x + 1) / (1-x)^5.

cp's OEIS Frontend

A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

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A214011 T(n,k) is the number of n X n nonnegative integer matrices with row and column i=1..n having sum <= i*k.

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A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

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A179824 Chromatic polynomial of the star graph on 4 vertices (claw graph) and the path graph on 4 vertices.

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A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

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A364922 a(n) is the square of the side length of a simplex whose n-dimensional inner hypervolume is equal to its (n-1)-dimensional surface hypervolume. As a result, the sequence starts at n=2.

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A242983 n/2 * (n^3 - 2*n^2 - 2*n + 5).

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A242983 n/2 * (n^3 - 2n^2 - 2n + 5).