A002817 -id:A002817 - cp's OEIS frontend

A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 6, 1, 0, 1, 6, 15, 20, 21, 8, 1, 0, 1, 7, 21, 35, 55, 39, 13, 1, 0, 1, 8, 28, 56, 120, 136, 92, 18, 1, 0, 1, 9, 36, 84, 231, 377, 430, 198, 30, 1, 0

Offset: 0

Robert A. Russell, Dec 18 2018

			Table begins with T(0,0):
  1 1  1    1     1      1       1        1        1         1         1 ...
  0 1  2    3     4      5       6        7        8         9        10 ...
  0 1  3    6    10     15      21       28       36        45        55 ...
  0 1  4   10    20     35      56       84      120       165       220 ...
  0 1  6   21    55    120     231      406      666      1035      1540 ...
  0 1  8   39   136    377     888     1855     3536      6273     10504 ...
  0 1 13   92   430   1505    4291    10528    23052     46185     86185 ...
  0 1 18  198  1300   5895   20646    60028   151848    344925    719290 ...
  0 1 30  498  4435  25395  107331   365260  1058058   2707245   6278140 ...
  0 1 46 1219 15084 110085  563786  2250311  7472984  21552969  55605670 ...
  0 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022 ...
For T(3,3)=10, the unoriented cycles are 9 achiral (AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC) and 1 chiral pair (ABC-ACB).

Index entries for sequences related to bracelets

Cf. A075195 (oriented), A293496(chiral), A284855 (achiral).
Cf. A051137 (ascending antidiagonals).
Columns 0-6 are A000007, A000012, A000029, A027671, A032275, A032276, and A056341.
Rows 0-7 are A000012, A001477, A000217, A000292, A002817, A060446, A027670, and A060532.
Main diagonal gives A081721.

Table[If[k>0, DivisorSum[k, EulerPhi[#](n-k)^(k/#)&]/(2k) + ((n-k)^Floor[(k+1)/2]+(n-k)^Ceiling[(k+1)/2])/4, 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d|n} phi(d) * k^(n/d).
T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.
T(n,k) = A075195(n,k) - A293496(n,k) = A293496(n,k) + A284855(n,k).
Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.
O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

A334358 Irregular triangle read by rows: row n gives scaled coefficients of the chromatic polynomial corresponding to colorings of the n-hypercube graph up to automorphism, highest powers first, 0 <= k <= 2^n.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, 3, -2, 0, 1, -12, 72, -256, 579, -812, 644, -216, 0, 1, -32, 496, -4936, 35276, -191840, 820328, -2808636, 7759343, -17276144, 30675244, -42494732, 44214736, -32375904, 14772272, -3125472, 0, 1, -80, 3160, -82080, 1575420, -23805776, 294640000

Offset: 0

Andrew Howroyd, Apr 24 2020

The polynomials are scaled by a factor of n!*2^n to ensure integer coefficients. When evaluated at x = k, they give the number of non-isomorphic k-colorings of the n-hypercube graph under the automorphism group of the graph. The size of the automorphism group is n!*2^n. Colors may not be interchanged.

			Triangle begins:
  0 | 1, 0;
  1 | 1, -1, 0;
  2 | 1, -2, 3, -2, 0;
  3 | 1, -12, 72, -256, 579, -812, 644, -216, 0;
  ...
The corresponding polynomials are:
  x;
  (x^2 - x)/(1!*2^1);
  (x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2);
  (x^8 - 12*x^7 + 72*x^6 - 256*x^5 + 579*x^4 - 812*x^3 + 644*x^2 - 216*x)/(3!*2^3);
  ...
The polynomial (x^4 - 2*x^3 + 3*x^2 - 2*x)/(2!*2^2) gives A002817 when evaluated at integer values of x.

Andrew Howroyd, Table of n, a(n) for n = 0..68 (rows 0..5)
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Eric Weisstein's World of Mathematics, Hypercube Graph

Cf. A002817, A334159, A334248, A334356, A334357.

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

Original entry on oeis.org

1, 120, 6210, 153040, 2224955, 22069251, 164176640, 976395820, 4855258305, 20856798285, 79315936751, 272095118010, 854560160105, 2486299719645, 6765755480415, 17356306529251, 42250330784180, 98137852369965

Offset: 0

N. J. A. Sloane

Number of 5 X 5 stochastic matrices of integers.

D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, p. 234.

T. D. Noe, Table of n, a(n) for n = 0..1000
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A002817, A001496, A019298.
Cf. A001496, A005466, A058391.
Row n=5 of A257493.

CoefficientList[Series[(1+103x+4306x^2+63110x^3+388615x^4+1115068x^5+ 1575669x^6+1115068x^7+388615x^8+63110x^9+4306x^10+103x^11+x^12)/ (1-x)^17,{x,0,30}],x] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: (1 + 103*x + 4306*x^2 + 63110*x^3 + 388615*x^4 + 1115068*x^5 + 1575669*x^6 + 1115068*x^7 + 388615*x^8 + 63110*x^9 + 4306*x^10 + 103*x^11 + x^12)/(1-x)^17.

a(n) = Sum_{j=0..6} A005466(j) * binomial(4+j+n, 4+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Vladeta Jovovic, Feb 06 2000

A005045 Number of restricted 3 X 3 matrices with row and column sums n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 17, 25, 37, 51, 70, 92, 121, 153, 194, 240, 296, 358, 433, 515, 612, 718, 841, 975, 1129, 1295, 1484, 1688, 1917, 2163, 2438, 2732, 3058, 3406, 3789, 4197, 4644, 5118, 5635, 6183, 6777, 7405, 8084, 8800, 9571, 10383, 11254

Offset: 0

N. J. A. Sloane

More precisely, consider 3 X 3 matrices with entries chosen from {0, 1, ..., n-1}, in which each row and column sums to n, where n >= 2. Then a(n) is the number of equivalence classes of such matrices under permutions of rows and columns and transpositions.

			a(2) = 1:
  110
  101
  011
a(3) = 3:
  111 210 210
  111 102 111
  111 021 012

E. J. Morgan, On 3 X 3 matrices with constant row and column sum, Abstract 763-05-13, Notices Amer. Math. Soc., 26 (1979), page A-27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler, Table of n, a(n) for n = 0..1000
E. J. Billington (née Morgan) and N. J. A. Sloane, Correspondence, 1978-1991.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
R. J. Mathar, OEIS A005045 [Proof of g.f. for 3 of the 12 cases]
E. J. Morgan, Construction of Block Designs and Related Results, Ph.D. Dissertation, Univ. Queensland, 1978; Bull. Austral. Math. Soc., Volume 19, Issue 1 August 1978, pp. 139-140.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Cf. A002817 for another version.

A005045:=-z**2*(-z**5+z**6-z**3+z+1)/((z**2+1)*(z**2+z+1)*(z+1)**2*(z-1)**5); # conjectured by Simon Plouffe in his 1992 dissertation; see formula lines here for the proof of correctness

a[n_] := Block[{k = Floor[(n + 2)/3]}, Sum[Sum[Sum[If[m + r == i/2, Floor[(n - 2*i + m + 2)/2], n - 2*i + m + 1], {r, 0, Floor[i/2 - m]}], {m, Max[2*i - n, 0], Floor[i/2]}], {i, 1, n - k}]]; Table[a[n], {n, 0, 46}] (* Peter Pein, May 13 2008 *)
LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{0,0,1,3,6,10,17,25,37,51,70},50] (* Harvey P. Dale, Nov 15 2018 *)

A005045(n)={sum( i=1,n-(n+2)\3, sum( m=max(0,2*i-n),i\2, sum( r=0,i\2-m, if( m+r!=i/2, n-2*i+m+1, (n-2*i+m+2)\2))))} \\ M. F. Hasler, Version 1, May 13 2008

A005045(n)={sum( i=1,(2*n)\3, sum( m=max(0,2*i-n),i\2, (n-2*i+m+1)*((i+1)\2-m)+(i%2==0)*(n-2*i+m+2)\2))} \\ M. F. Hasler, Version 2, much faster, May 13 2008

concat(vector(2), Vec(x^2*(1 + x - x^3 - x^5 + x^6) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Apr 22 2017

Let n = 3k, 3k-1 or 3k-2 according as n == 0, 2 or 1 mod 3, for n >= 3. Then a(n) = Sum_{i=1..n-k} Sum_{m=max(0,2i-n)..floor(i/2)} Sum_{r=0..floor(i/2)-m} c(i,m,r), where c(i,m,r) = n-2i+m+1 when m+r != i/2, or = floor((n-2i+m+2)/2) when m+r = i/2. [Typos corrected by Peter Pein, May 13 2008]

G.f.: -x^2*(-x^5+x^6-x^3+x+1)/((x^2+1)*(x^2+x+1)*(x+1)^2*(x-1)^5). This was conjectured by Simon Plouffe in his 1992 dissertation and is now known to be correct, although it may be that all the details of the proof have not been written down. See the Mathar link for details.

Edited by N. J. A. Sloane, May 12 2008, May 13 2008

More terms from Peter Pein, May 13 2008

A095693 Triangle read by rows: T(n,k) is the number of multigraphs without loops on n labeled nodes with k edges and maximum degree 2.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 6, 1, 1, 6, 21, 22, 6, 1, 10, 55, 130, 130, 22, 1, 15, 120, 485, 1005, 822, 130, 1, 21, 231, 1400, 4830, 8547, 6202, 822, 1, 28, 406, 3416, 17465, 52052, 81676, 52552, 6202, 1, 36, 666, 7392, 52101, 230832, 610932, 859932, 499194, 52552

Offset: 0

Nicholas S. Horne (nickhorne(AT)cox.net), Jul 06 2004

Sum of the each row of the triangle corresponds to sequence A000985. The diagonal of the triangular array T(n,1) represents the triangular numbers (A000217). The T(n,2) diagonal represents the doubly triangular numbers (A002817).

Number of symmetric n X n matrices with nonnegative integer entries and all row sums 2 and trace 2*(n-k). - Andrew Howroyd, Nov 07 2019

			Triangle begins:
  1;
  1,  0;
  1,  1,   1;
  1,  3,   6,    1;
  1,  6,  21,   22,    6;
  1, 10,  55,  130,  130,   22;
  1, 15, 120,  485, 1005,  822,  130;
  1, 21, 231, 1400, 4830, 8547, 6202, 822;
  ...
T(3,2)=6 since there are six ways that a multigraph with 3 nodes can be constructed with 2 edges such that no vertex has degree greater than two.

Horne, Nicholas S. "Analysis of Viable Network Configurations from a Combinatorial, Graphical and Algebraic Perspective." Diss. Providence College, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A000985.
Main diagonal is A002137.
Columns include A000217, A002817.

T(n)={my(v=Vec(serlaplace(sqrt(1/(1-x*y) + O(x*x^n))*exp(x + (x^2*y/(1-x*y) - x*y)/2 + x^2*y^2/4 + O(x*x^n))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 07 2019

E.g.f.: sqrt(1/(1-x*y)) * exp(x + (x^2*y/(1-x*y) - x*y)/2 + x^2*y^2/4). - Andrew Howroyd, Nov 07 2019

Definition clarified and terms a(37) and beyond from Andrew Howroyd, Nov 07 2019

A003439 Number of 6 X 6 stochastic matrices of integers: all rows and columns sum to n.

Original entry on oeis.org

1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, 8575979362560, 94459713879600, 842286559093240, 6292583664553881, 40447642842118656, 228438173705550566, 1152877640765297760, 5271278793334883190, 22085628572718605376, 85604721304213863531

Offset: 0

N. J. A. Sloane

nonn

Matthias Beck and Dennis Pixton, The Ehrhart Polynomial of the Birkhoff Polytope, Discrete & Computational Geometry, 30(4)(2003), 623-637.
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Row n=6 of A257493.

Cf. A005467.

a(n) = Sum_{j=0..10} A005467(j) * binomial(5+j+n, 5+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Melissa Erdmann (merdmann(AT)nebrwesleyan.edu), May 07 2009
Offset changed to 0 by Alois P. Heinz, Apr 26 2015
Name clarified by Charles R Greathouse IV, Mar 03 2018

A070333 Expansion of (1 + x)(1 - x + x^2)/((1 - x)^4(1 + x + x^2)).

Original entry on oeis.org

1, 3, 6, 12, 21, 33, 50, 72, 99, 133, 174, 222, 279, 345, 420, 506, 603, 711, 832, 966, 1113, 1275, 1452, 1644, 1853, 2079, 2322, 2584, 2865, 3165, 3486, 3828, 4191, 4577, 4986, 5418, 5875, 6357, 6864, 7398, 7959, 8547, 9164, 9810, 10485

Offset: 0

N. J. A. Sloane, May 11 2002

a(n) is the number of 3 X 3 matrices with nonnegative integer entries such that every row sum, column sum and the trace of the matrix is n. - Sharon Sela (sharonsela(AT)hotmail.com), May 20 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Cf. A059827, A001844, A002817.

[Round((2*n+3)*(n^2+3*n+6)/18): n in [0..50]]; // Vincenzo Librandi, Jun 25 2011

A049347 := proc(n) op(1+(n mod 3),[1,-1,0]) ; end proc:
A070333 := proc(n) 1+7*n/6+n^2/2+n^3/9+2*A049347(n-1)/9 ; end proc: # R. J. Mathar, Dec 03 2010

CoefficientList[ Series[(1 + x^3)/(1 - 3*x + 3*x^2 - 2*x^3 + 3*x^4 - 3*x^5 + x^6), {x, 0, 45}], x]

Vec((1+x)*(1-x+x^2)/((1-x)^4*(1+x+x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = 1 + 7*n/6 + n^2/2 + n^3/9 + 2*A049347(n-1)/9. - R. J. Mathar, Dec 03 2010
From Mircea Merca, Dec 03 2010: (Start)
a(n) = round((2*n + 3)*(n^2 + 3*n + 6)/18).
a(n) = floor((n + 2)*(2*n^2 + 5*n + 11)/18).
a(n) = ceiling((n + 1)*(2*n^2 + 7*n + 14)/18).
a(n) = round((n + 1)*(2*n^2 + 7*n + 14)/18).
a(n) = a(n-3) + n^2 + 2 for n > 2. (End)
E.g.f.: exp(x)*(1 + x*(32 + x*(15 + 2*x))/18) + 4*exp(-x/2)*sin(sqrt(3)*x/2)/(9*sqrt(3)). - Stefano Spezia, Oct 28 2022

A217338 Number of inequivalent ways to color a 4 X 4 checkerboard using at most n colors allowing rotations and reflections.

Original entry on oeis.org

0, 1, 8548, 5398083, 537157696, 19076074375, 352654485156, 4154189102413, 35184646816768, 231628411446741, 1250002537502500, 5743722797690911, 23110548002468928, 83177110918426603, 272244240093265636, 821051189587805625, 2305843285702230016, 6082649491072763593

Offset: 0

Geoffrey Critzer, Oct 01 2012

Cycle index of symmetry group: (s(1)^16 + 2*s(4)^4 + 3*s(2)^8 + 2*s(2)^6*s(1)^4)/8.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Row n=4 of A343097.

Cf. A002817, A217331.

Table[(n^16+2n^4+3n^8+2n^10)/8, {n,0,20}]

a(n) = (n^16 + 2*n^4 + 3*n^8 + 2*n^10)/8; \\ Indranil Ghosh, Feb 27 2017

def A217338(n): return (n**16 + 2*n**4 + 3*n**8 + 2*n**10)/8 # Indranil Ghosh, Feb 27 2017

a(n) = (n^16 + 2*n^4 + 3*n^8 + 2*n^10)/8.

G.f.: -x*(x +1)*(x^14 +8530*x^13 +5244373*x^12 +441307760*x^11 +10231414811*x^10 +87532894238*x^9 +313403397135*x^8 +484445834304*x^7 +313403397135*x^6 +87532894238*x^5 +10231414811*x^4 +441307760*x^3 +5244373*x^2 +8530*x +1)/(x -1)^17. [Colin Barker, Oct 04 2012]

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.

Original entry on oeis.org

0, 1, 14, 61, 175, 400, 791, 1414, 2346, 3675, 5500, 7931, 11089, 15106, 20125, 26300, 33796, 42789, 53466, 66025, 80675, 97636, 117139, 139426, 164750, 193375, 225576, 261639, 301861, 346550, 396025, 450616, 510664, 576521, 648550, 727125, 812631, 905464, 1006031

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of centered 11-gonal (or hendecagonal) pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004467.

Cf. similar sequences provided by the partial sums of centered k-gonal pyramidal numbers: A006522 (k=1), A006007 (k=2), A002817 (k=3), A006325 (k=4), A006322 (k=5), A000537 (k=6), A006323 (k=7), A006324 (k=8), A236770 (k=9), A264853 (k=10), this sequence (k=11), A062392 (k=12), A264888 (k=13).

[n*(n+1)*(11*n^2+11*n-10)/24: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (11 n^2 + 11 n - 10)/24, {n, 0, 50}]

a(n)=n*(n+1)*(11*n^2+11*n-10)/24 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 9*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004467(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

A290945 Triangular Carmichael numbers.

Original entry on oeis.org

561, 8911, 10585, 41041, 115921, 314821, 334153, 6313681, 8134561, 14913991, 32914441, 60957361, 67902031, 135556345, 289766701, 321197185, 329769721, 368113411, 471905281, 765245881, 842202361, 962442001, 1507746241, 2489462641, 2588653081, 3104207821

Offset: 1

Amiram Eldar, Aug 14 2017

nonn

Intersection of A000217 and A002997.
The least triangular Carmichael numbers with the number of prime factors = 3, 4, 5, 6, 7, ... are: 561, 41041, 765245881, 321197185, 1583892181303201, ...
The number of terms below 10^k for k = 3, 4, ... are: 1, 2, 4, 7, 9, 13, 22, 32, 53, 77, 137, 211, 358, 545, 879, 1423, ...
Jonathan Vos Post discovered in 2004 that a(21) = 842202361 = A000217(41041) = A002817(286) is also a doubly triangular Carmichael number. The next number with this property is a(1108) = 292800629576356021 = A000217(765245881) = A002817(39121) (41041 and 765245881 are triangular Carmichael numbers that are also indices of triangular numbers that are also Carmichael numbers).

			8911 = A000217(133) = A002997(7) therefore 8911 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..8920 (terms below 10^22 calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Jonathan Vos Post, Table of Figurate Numbers, Sorted, Through 10,000. [Wayback Machine link]
Index entries for sequences related to Carmichael numbers.

Cf. A000217, A002817, A002997.

seqQ[n_]:=IntegerQ[Sqrt[8n+1]] && !PrimeQ[n] && (Mod[n, CarmichaelLambda[n]] == 1); Select[Range[10^6], seqQ]

cp's OEIS Frontend

A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

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A334358 Irregular triangle read by rows: row n gives scaled coefficients of the chromatic polynomial corresponding to colorings of the n-hypercube graph up to automorphism, highest powers first, 0 <= k <= 2^n.

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A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

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A005045 Number of restricted 3 X 3 matrices with row and column sums n.

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Maple

Mathematica

PARI

PARI

PARI

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A095693 Triangle read by rows: T(n,k) is the number of multigraphs without loops on n labeled nodes with k edges and maximum degree 2.

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A003439 Number of 6 X 6 stochastic matrices of integers: all rows and columns sum to n.

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A070333 Expansion of (1 + x)*(1 - x + x^2)/((1 - x)^4*(1 + x + x^2)).

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Magma

Maple

A070333 Expansion of (1 + x)(1 - x + x^2)/((1 - x)^4(1 + x + x^2)).

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.