A047996 -id:A047996 - cp's OEIS frontend

A215251 Product of terms in n-th row of A037306.

1, 1, 1, 2, 4, 36, 225, 7840, 313600, 45302400, 8930250000, 8373836401920, 9001015156742400, 41813367543204433176, 325385777102562972821025, 8270190445766978650521600000, 377177413291384771899817984000000, 62187743659065606074696974956949929984

Offset: 1

N. J. A. Sloane, Sep 05 2012

Also products of terms in rows of A047996.

Alois P. Heinz, Table of n, a(n) for n = 1..50
R. Bekes, J. Pedersen and B. Shao, Mad tea party cyclic partitions, College Math. J., 43 (2012), 24-36.

Cf. A037306, A008965, A047996.

with (numtheory):
a:= n-> mul (add(phi(d)*binomial(n/d, k/d),
                     d=divisors(igcd(n, k))), k=0..n)/n^(n+1):
seq (a(n), n=1..20);  # Alois P. Heinz, Sep 06 2012

t[n_, k_] := Total[EulerPhi[#] * Binomial[n/#, k/#]& /@ Divisors[GCD[n, k]]]/n; Table[Times @@ Table[t[n, k], {k, 1, n}], {n, 1, 18}] (* Jean-François Alcover, Mar 07 2014 *)

A032195 Number of necklaces with 10 black beads and n-10 white beads.

Original entry on oeis.org

1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 16796, 29414, 49742, 81752, 130752, 204347, 312455, 468754, 690690, 1001603, 1430715, 2016144, 2804880, 3856892, 5245128, 7060984, 9414328, 12440668, 16301164

Offset: 10

Christian G. Bower

nonn

The g.f. is Z(C_10,x)/x^10, the 10-variate cycle index polynomial for the cyclic group C_10, with substitution x[i]->1/(1-x^i), i=1,...,10. By Polya enumeration, a(n+10) is the number of cyclically inequivalent 10-necklaces whose 10 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_10,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).

Column k=10 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194.

k = 10; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of 1, 1, 1, 1...
G.f.: (x^10)*(1-3*x+4*x^2+12*x^3-8*x^4-x^5+31*x^6-4*x^8+16*x^9 +11*x^10 +3*x^11+8*x^12+4*x^13+4*x^14+x^15+x^16) /((1-x)^4*(1-x^2)^4 *(1-x^5)*(1-x^10)).
G.f.: (1/10)*x^10*(1/(1 - x)^10 + 1/(1 - x^2)^5 + 4/(1 - x^5)^2 + 4/(1 - x^10)^1). - Herbert Kociemba, Oct 22 2016

A032196 Number of necklaces with 11 black beads and n-11 white beads.

Original entry on oeis.org

1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32066, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864750, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796

Offset: 11

Christian G. Bower

nonn

The g.f. is Z(C_11,x)/x^11, the 11-variate cycle index polynomial for the cyclic group C_11, with substitution x[i]->1/(1-x^i), i=1..11. By Polya enumeration, a(n+11) is the number of cyclically inequivalent 11-necklaces whose 11 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_11,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).

Column k=11 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194, A032195.

k = 11; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
DeleteCases[CoefficientList[Series[(x^11) (1 - 9 x + 41 x^2 - 109 x^3 + 191 x^4 - 229 x^5 + 191 x^6 - 109 x^7 + 41 x^8 - 9 x^9 + x^10)/((1 - x)^10 (1 - x^11)), {x, 0, 39}], x], 0] (* Michael De Vlieger, Oct 10 2016 *)

"CIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of 1, 1, 1, 1...
G.f.: (x^11) * (1 - 9*x + 41*x^2 - 109*x^3 + 191*x^4 - 229*x^5 + 191*x^6 - 109*x^7 + 41*x^8 - 9*x^9 + x^10) / ((1-x)^10 * (1-x^11)).
a(n) = ceiling(binomial(n, 11)/n) (conjecture Wolfdieter Lang).
From Herbert Kociemba, Oct 11 2016: (Start)
This conjecture indeed is true.
Sketch of proof:
There are binomial(n,11) ways to place the 11 black beads in the necklace with n beads. If n is not divisible by 11 there are no necklaces with a rotational symmetry. So exactly n necklaces are equivalent up to rotation and there are binomial(n,11)/n = ceiling(binomial(n,11)/n) equivalence classes.
If n is divisible by 11 the only way to get a necklace with rotational symmetry is to space out the 11 black beads evenly. There are n/11 ways to do this and all ways are equivalent up to rotation. So there are  binomial(n,11) - n/11 unsymmetric necklaces which give binomial(n,11)/n - 1/11 equivalence classes. If we add the single symmetric equivalence class we get Binomial(n,11)/n - 1/11 + 1 which also is ceiling(binomial(n,11)/n). (End)
G.f.: (10/(1 - x^11) + 1/(1 - x)^11)*x^11/11. - Herbert Kociemba, Oct 16 2016

A032197 Number of necklaces with 12 black beads and n-12 white beads.

Original entry on oeis.org

1, 1, 7, 31, 116, 364, 1038, 2652, 6310, 14000, 29414, 58786, 112720, 208012, 371516, 643856, 1086601, 1789515, 2883289, 4552275, 7056280, 10752060, 16128424, 23841480, 34769374, 50067108, 71250060, 100276894, 139672312

Offset: 12

Christian G. Bower

nonn

The g.f. is Z(C_12,x)/x^12, the 12-variate cycle index polynomial for the cyclic group C_12, with substitution x[i]->1/(1-x^i), i=1,...,12. Therefore by Polya enumeration a(n+12) is the number of cyclically inequivalent 12-necklaces whose 12 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_12,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).

Column k=12 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192-A032196.

k = 12; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 12 ]" (necklace, indistinct, unlabeled, 12 parts) transform of 1, 1, 1, 1...
G.f.: (x^12)*(1-3*x+7*x^2+9*x^3+18*x^4+38*x^5+72*x^6+92*x^7+168*x^8+160*x^9+238*x^10+230*x^11+296*x^12+234*x^13+330*x^14+248*x^15+284*x^16+238*x^17+230*x^18+166*x^19+172*x^20+78*x^21+80*x^22+38*x^23+21*x^24+7*x^25+3*x^26+x^27) /((1+x)*(1-x)*(1-x^2)*(1-x^3)*(1-x)^5*(1+x+x^2)*(1-x^4)^2*(1-x^6)*(1-x^12)). - Wolfdieter Lang, Feb 15 2005 (see comment)
G.f.: 1/12 x^12 ((1 - x)^-12 + (1 - x^2)^-6 + 2 (1 - x^3)^-4 + 2 (1 - x^4)^-3 + 2 (1 - x^6)^-2 + 4 (1 - x^12)^-1). - Herbert Kociemba, Oct 22 2016

A211352 Refined triangle A211356: T(n,k) is the number of partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 4, 9, 3, 10, 1, 5, 3, 3, 1, 1, 3, 5, 15, 5, 30, 3, 15, 15, 10, 1, 15, 3, 5, 1, 1, 4, 7, 29, 10, 70, 7, 56, 54, 37, 4, 105, 21, 35, 1, 18, 29, 37, 4, 7, 7, 1

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,7,12,43... (A084423)
Row maxima: 1,1,1,2,3,10,30,105,420,1268,6300...
Distinct entries per row: 1,1,1,2,3,6,6,13,17,25,25...
Rightmost columns are those from the triangle of circular binomial coefficients A047996 without the second column (i.e.triangle A037306).

Tilman Piesk, Rows n=1..11 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211356, A000041, A084423, A047996, A037306.

A211353 Refined triangle A211357: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 4, 6, 3, 5, 1, 2, 1, 1, 1, 1, 3, 5, 10, 5, 15, 3, 5, 6, 3, 1, 3, 1, 1, 1, 1, 4, 7, 19, 10, 35, 7, 19, 21, 12, 4, 21, 7, 7, 1, 3, 4, 4, 1, 1, 1, 1, 1, 4, 10, 28, 14, 70, 14, 48, 56, 28, 10

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,6,10,28... (A054357)
Row maxima: 1,1,1,2,2,6,15,35,84,252,630,1542...
Distinct entries per row: 1,1,1,2,2,6,6,9,11,17,17,30...
Rightmost columns are those from the triangle of circular binomial coefficients A047996 without the second column (i.e.triangle A037306).

Tilman Piesk, Rows n=1..12 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211357, A000041, A054357, A047996, A037306.

A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 3, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 9, 7, 4, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 5, 12, 21, 26, 21, 12, 5, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 6, 19, 42, 66, 77, 66, 42, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Jan 22 2019

For n >= 1, T(n,k) is the number of nodes in n-dimensional space for Mysovskikh's cubature formula which is exact for any polynomial of degree k of n variables.

			Array begins:
  1, 1, 1,  1,  1,   1,   1,    1,    1,    1, ...
  1, 1, 2,  2,  3,   3,   4,    4,    5,    5, ...
  1, 1, 2,  4,  5,   7,  10,   12,   15,   19, ...
  1, 1, 3,  5,  9,  14,  21,   30,   42,   55, ...
  1, 1, 3,  7, 14,  26,  42,   66,   99,  143, ...
  1, 1, 4, 10, 21,  42,  77,  132,  215,  334, ...
  1, 1, 4, 12, 30,  66, 132,  246,  429,  715, ...
  1, 1, 5, 15, 42,  99, 215,  429,  805, 1430, ...
  1, 1, 5, 19, 55, 143, 334,  715, 1430, 2702, ...
  1, 1, 6, 22, 72, 201, 501, 1144, 2431, 4862, ...
  ...
As triangular array, this begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  4,  3,  1,  1;
  1, 3,  5,  5,  3,  1,  1;
  1, 4,  7,  9,  7,  4,  1,  1;
  1, 4, 10, 14, 14, 10,  4,  1, 1;
  1, 5, 12, 21, 26, 21, 12,  5, 1, 1;
  1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
  ...

Ronald Cools, Encyclopaedia of Cubature Formulas
Ivan P. Mysovskikh, On the construction of cubature formulae for very simple domains, USSR Computational Mathematics and Mathematical Physics, Volume 4, Issue 1, 1964, 1-17.

Main diagonal: A000108.

Cf. A007318, A037306, A046854, A047996, A065941, A241926, A267632.

b(n, k) := (n + k)!/((n + 1)!*k!)$
T(n, k) := if integerp(b(n, k)) then b(n, k) else floor(b(n, k)) + 1$
create_list(T(k, n - k), n, 0, 15, k, 0, n);

A273090 Rectangular array A read by upward antidiagonals in which the entry A(n,k) in row k and column n gives the number of families of symmetric radially generated monohedral tilings of the disk (each tiling contains 2(2n+1)*k congruent tiles), k >= 1, n >= 1.

Original entry on oeis.org

2, 62, 2, 116, 1532, 2, 200, 6402, 50830, 2, 318, 19884, 446930, 1855110, 2, 476, 51128, 2460462, 34121322, 71292624, 2, 682, 115188, 10106370, 332112068, 2741227176, 2833906726, 2, 946, 235180, 33905948, 2177193500, 47162138964

Offset: 1

L. Edson Jeffery, May 14 2016

Enumeration is equivalent to counting beaded necklaces of a certain class (see A047996). For details and definitions, see the arXiv preprint by Haddley and Worsley.

			Array begins:
.    2       2         2            2              2                 2
.   62    1532     50830      1855110       71292624        2833906726
.  116    6402    446930     34121322     2741227176      227759341712
.  200   19884   2460462    332112068    47162138964     6926365932512
.  318   51128  10106370   2177193500   493416845604   115646287581042
.  476  115188  33905948  10874491594  3668999040616  1280224897307324

Joel Haddley, Stephen Worsley, Infinite families of monohedral disk tilings, arXiv preprint arXiv:1512.03794v2 [math.MG], 2016.

Cf. A047996.

a[1, n_] := 2; a[k_, n_] := 2*(1 + Sum[(1/i)*Sum[EulerPhi[d]*Binomial[(2*(2*n + 1) - i)*k/d + i/d - 1, i/d - 1], {d, Divisors[GCD[i, (2*(2*n + 1) - i)*k]]}], {i, 2*(2*n + 1)}]);
(* Array: *)
Grid[Table[a[k, n], {k, 6}, {n, 6}]]
(* Or array antidiagonals flattened: *)
Flatten[Table[a[k - n + 1, n], {k, 7}, {n, k}]]

A(1,n) = 2, A(k,n) = 2*Sum_{i=0..2*(2*n+1)}Sum_{d | i, d | (2*(2*n+1)-i)*k} (phi(d)/i)*binomial((2*(2*n+1)-i)*k/d+i/d-1, i/d-1), k >= 2, n >= 1 [Haddley, Worsley, Proposition 5.1].

cp's OEIS Frontend

A215251 Product of terms in n-th row of A037306.

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A032195 Number of necklaces with 10 black beads and n-10 white beads.

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A032196 Number of necklaces with 11 black beads and n-11 white beads.

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A032197 Number of necklaces with 12 black beads and n-12 white beads.

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A211352 Refined triangle A211356: T(n,k) is the number of partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

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A211353 Refined triangle A211357: T(n,k) is the number of noncrossing partitions up to rotation of an n-set that are of type k (k-th integer partition, defined by A194602).

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A322596 Square array read by descending antidiagonals (n >= 0, k >= 0): let b(n,k) = (n+k)!/((n+1)!*k!); then T(n,k) = b(n,k) if b(n,k) is an integer, and T(n,k) = floor(b(n,k)) + 1 otherwise.

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Maxima

A273090 Rectangular array A read by upward antidiagonals in which the entry A(n,k) in row k and column n gives the number of families of symmetric radially generated monohedral tilings of the disk (each tiling contains 2*(2*n+1)*k congruent tiles), k >= 1, n >= 1.

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A273090 Rectangular array A read by upward antidiagonals in which the entry A(n,k) in row k and column n gives the number of families of symmetric radially generated monohedral tilings of the disk (each tiling contains 2(2n+1)*k congruent tiles), k >= 1, n >= 1.