A000048 -id:A000048 - cp's OEIS frontend

A325015 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

1, 2, 1, 3, 6, 1, 4, 18, 21, 1, 5, 40, 201, 308, 1, 6, 75, 1076, 34128, 180342, 1, 7, 126, 4025, 1056576, 2945136213, 366975285216, 1, 8, 196, 11901, 15303750, 2932338749408, 103863386269870076808, 10316179427644325573474464, 1

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. An achiral coloring is identical to its reflection.

Also the number of achiral colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with T(1,1):
1   2     3       4        5         6         7          8 ...
1   6    18      40       75       126       196        288 ...
1  21   201    1076     4025     11901     29841      66256 ...
1 308 34128 1056576 15303750 136236276 865711763 4296782848 ...
...
For T(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325012 (oriented), A325013 (unoriented), A325014 (chiral), A325019 (exactly k colors).
Other n-dimensional polytopes: A325001 (simplex), A325007 (orthotope).
Rows 1-2 are A000027, A002411.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[(CI1[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
T(n,k) = 2*A325013(n,k) - A325012(n,k) = A325012(n,k) - 2*A325014(n,k) = A325013(n,k) - A325014(n,k).
T(n,k) = Sum_{j=1..3*2^(n-2)} A325019(n,j) * binomial(k,j).

A325013 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 10, 21, 22, 1, 15, 55, 267, 402, 1, 21, 120, 1996, 132102, 1228158, 1, 28, 231, 10375, 11756666, 484086357207, 400507806843728, 1, 36, 406, 41406, 405385550, 4805323147589984, 74515759884862073604656433, 527471432057653004017274030725792, 1

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with A(1,1):
1   3      6       10        15         21          28           36 ...
1   6     21       55       120        231         406          666 ...
1  22    267     1996     10375      41406      135877       384112 ...
1 402 132102 11756666 405385550 7416923886 86986719477 735192450952 ...
For A(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325012 (oriented), A325014 (chiral), A325015 (achiral), A325017 (exactly k colors).
Other n-dimensional polytopes: A325000 (simplex), A325005 (orthotope).
Rows 1-4 are A000217, A002817, A128766, A128767; column 2 is A000616.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[((CI0[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = A325012(n,k) - A325014(n,k) = (A325012(n,k) + A325015(n,k)) / 2 = A325014(n,k) + A325015(n,k).
A(n,k) = Sum_{j=1..2^n} A325017(n,j) * binomial(k,j).

A325016 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Original entry on oeis.org

1, 2, 1, 4, 9, 6, 1, 21, 267, 1718, 5250, 7980, 5880, 1680, 1, 494, 228591, 21539424, 685479375, 10257064650, 86151316860, 449772354360, 1551283253100, 3661969537800, 6015983173200, 6878457986400, 5371454088000, 2733402672000, 817296480000, 108972864000

Offset: 1

Robert A. Russell, May 28 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

Also the number of oriented colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
  1  2
  1  4   9    6
  1 21 267 1718 5250 7980 5880 1680
For T(2,2)=4, two squares have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..510, rows 1..8, flattened.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325017 (unoriented), A325018 (chiral), A325019 (achiral), A325012 (up to k colors).
Other n-dimensional polytopes: A325002 (simplex), A325008 (orthotope).
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[(CI0[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325012 *)
Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten

A325012(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325017(n,k) + A325018(n,k) = 2*A325017(n,k) - A325019(n,k) = 2*A325018(n,k) + A325019(n,k).

A325017 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Original entry on oeis.org

1, 1, 1, 4, 6, 3, 1, 20, 204, 1056, 2850, 4080, 2940, 840, 1, 400, 130899, 11230666, 347919225, 5158324560, 43174480650, 225086553300, 775894225050, 1831178115900, 3008073915000, 3439243962000, 2685727044000, 1366701336000, 408648240000, 54486432000

Offset: 1

Robert A. Russell, Jun 09 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
1  1
1  4   6    3
1 20 204 1056 2850 4080 2940 840
For T(2,2)=4, two squares have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..510, rows 1..8, flattened.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A325016 (oriented), A325018 (chiral), A325019 (achiral), A325013 (up to k colors).
Other n-dimensional polytopes: A007318(n,k-1) (simplex), A325009 (orthotope).
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ],MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[((CI0[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325013 *)
Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten

A325013(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325016(n,k) - A325018(n,k) = (A325016(n,k) + A325019(n,k)) / 2 = A325018(n,k) + A325019(n,k).

A325019 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Original entry on oeis.org

1, 0, 1, 4, 3, 0, 1, 19, 141, 394, 450, 180, 0, 0, 1, 306, 33207, 921908, 10359075, 59584470, 197644440, 400752240, 505197000, 386694000, 164656800, 29937600, 0, 0, 0, 0

Offset: 1

Robert A. Russell, Jun 09 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. An achiral coloring is identical to its reflection. The last 2^(n-2) columns of row n are zero; there are no achiral colorings with that many colors.

Also the number of achiral colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
1  0
1  4   3   0
1 19 141 394 450 180 0 0
For T(2,3)=3, each square has one of the three colors on two opposite edges.

Robert A. Russell, Table of n, a(n) for n = 1..510, rows 1..8, flattened.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A325016 (oriented), A325017 (unoriented), A325018 (chiral), A325015 (up to k colors).
Other n-dimensional polytopes: A325003 (simplex), A325011 (orthotope).
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ],MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n,MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even permutation *)
CI1[{n_Integer}] := CI1[{{n}}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd permutation *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]] == s[[i - 1, 1]], s[[i - 1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[(CI1[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325012 *)
Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten

A325015(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = 2*A325017(n,k) - A325016(n,k) = A325016(n,k) - 2*A325018(n,k) = A325017(n,k) - A325018(n,k).

A002075 Number of equivalence classes with primitive period n of base 3 necklaces, where necklaces are equivalent under rotation and permutation of symbols.

Original entry on oeis.org

1, 1, 2, 4, 8, 22, 52, 140, 366, 992, 2684, 7404, 20440, 56992, 159440, 448540, 1266080, 3587610, 10195276, 29057520, 83018728, 237737984, 682196948, 1961323740, 5648590728, 16294032160, 47071589778, 136171440600

Offset: 1

N. J. A. Sloane

N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for sequences related to necklaces

Cf. A000013, A000048, A002076.

Reference gives formula.

Sequence A002076 can be found as follows: Let F3(n) = this sequence, F3*(n) = function from A002076. Then F3*(n) = Sum_{ d divides n } F3(d).

Better description and more terms from Mark Weston (mweston(AT)uvic.ca), Oct 07 2001

A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Original entry on oeis.org

0, 1, 0, 0, 3, 3, 0, 1, 63, 662, 2400, 3900, 2940, 840, 0, 94, 97692, 10308758, 337560150, 5098740090, 42976836210, 224685801060, 775389028050, 1830791421900, 3007909258200, 3439214024400, 2685727044000, 1366701336000, 408648240000, 54486432000

Offset: 1

Robert A. Russell, Jun 09 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. The chiral colorings of its facets come in pairs, each the reflection of the other.

Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
0 1
0 0  3   3
0 1 63 662 2400 3900 2940 840
For T(2,3)=3, each square has one of the three colors on two adjacent edges.

Robert A. Russell, Table of n, a(n) for n = 1..510, rows 1..8, flattened.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A325016 (oriented), A325017 (unoriented), A325019 (achiral), A325014 (up to k colors).
Other n-dimensional polytopes: A325010 (orthotope).
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ],MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[((CI0[#] - CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325014 *)
Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten

A325014(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325016(n,k) - A325017(n,k) = (A325016(n,k) - A325019(n,k)) / 2 = A325017(n,k) - A325019(n,k).

A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

Original entry on oeis.org

1, 2, 4, 6, 16, 30, 60, 126, 256, 504, 1020, 2046, 4080, 8190, 16380, 32730, 65536, 131070, 262080, 524286, 1048560, 2097018, 4194300, 8388606, 16776960, 33554400, 67108860, 134217216, 268435440, 536870910, 1073740740, 2147483646

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000048, A007727, A045662, A045664, A064355.

a[n_] := If[n==0, 1, 2n Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a /@ Range[0, 31] (* Jean-François Alcover, Sep 23 2019 *)

a(n)={if(n<1, n==0, sumdiv(n, d, if(d%2, moebius(d)*2^(n/d))))} \\ Andrew Howroyd, Sep 14 2019

from sympy import mobius, divisors
def A045663(n): return sum(mobius(d)<>(~n&n-1).bit_length(),generator=True)) if n else 1 # Chai Wah Wu, Jul 22 2024

a(n) = 2*n*A000048(n) = n*A064355(n) for n > 0.

a(n) = Sum{d|n, d odd} mu(d) * 2^(n/d) for n > 0. - Andrew Howroyd, Sep 14 2019

A053633 Triangular array T(n,k) giving coefficients in expansion of Product_{j=1..n} (1+x^j) mod x^(n+1)-1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 4, 3, 3, 3, 3, 6, 5, 5, 6, 5, 5, 10, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 30, 28, 28, 29, 28, 28, 29, 28, 28, 52, 51, 51, 51, 51, 52, 51, 51, 51, 51, 94, 93, 93, 93, 93, 93, 93, 93, 93, 93, 93, 172, 170, 170, 172, 170, 170, 172

Offset: 0

N. J. A. Sloane, Mar 22 2000

T(n,k) = number of binary vectors (x_1,...,x_n) satisfying Sum_{i=1..n} i*x_i = k (mod n+1) = size of Varshamov-Tenengolts code VT_k(n).

			Triangle begins:
  k  0    1    2    3    4    5    6    7    8    9
n
0    1;
1    1,   1;
2    2,   1,   1;
3    2,   2,   2,   2;
4    4,   3,   3,   3,   3;
5    6,   5,   5,   6,   5,   5;
6   10,   9,   9,   9,   9,   9,   9;
7   16,  16,  16,  16,  16,  16,  16,  16;
8   30,  28,  28,  29,  28,  28,  29,  28,  28;
9   52,  51,  51,  51,  51,  52,  51,  51,  51,  51;
    ...
[Edited by _Seiichi Manyama_, Mar 11 2018]

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.

Seiichi Manyama, Rows n = 0..139, flattened
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
N. J. A. Sloane, On single-deletion-correcting codes
Index entries for sequences related to subset sums modulo m
Index entries for sequences related to Gijswijt's sequence

Cf. A053632, A063776, A300328, A300628. Leading coefficients give A000016, next column gives A000048.

with(numtheory): A053633 := proc(n,k) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+(1/(2*n))*2^(n/d)*phi(d)*mobius(d/gcd(d,k))/phi(d/gcd(d,k)); fi; od; t1; end;

Flatten[ Table[ CoefficientList[ PolynomialMod[ Product[1+x^j, {j,1,n}], x^(n+1)-1], x], {n,0,11}]] (* Jean-François Alcover, May 04 2011 *)

The Maple code gives an explicit formula.

A054661 Number of monic irreducible polynomials over GF(4) with zero trace.

Original entry on oeis.org

1, 0, 5, 12, 51, 160, 585, 2016, 7280, 26112, 95325, 349180, 1290555, 4792320, 17895679, 67104768, 252645135, 954422560, 3616814565, 13743842916, 52357696365, 199911014400, 764877654105, 2932030307680, 11258999068416

Offset: 1

N. J. A. Sloane, Apr 18 2000

Also number of Lyndon words of length n with trace 0 over GF(4).

F. Ruskey, Number of monic irreducible polynomials over GF(q) with given trace
F. Ruskey, Number of Lyndon words over GF(q) with given trace

Cf. A000048, A051841, A046211, A046209, A054660, A074031, A074032, A074446, A074447.

From Andrey Zabolotskiy, Dec 17 2020: (Start)
a(n) = A074031(n) + 3 * A074032(n).
a(n) = A074446(n) + 3 * A074447(n). (End)

More terms from James Sellers, Apr 19 2000

cp's OEIS Frontend

A325015 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

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A325013 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

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A325016 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A325017 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A325019 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A002075 Number of equivalence classes with primitive period n of base 3 necklaces, where necklaces are equivalent under rotation and permutation of symbols.

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A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A045663 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to complement.

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