A035206 -id:A035206 - cp's OEIS frontend

A178888 Irregular table A035206(n,k)*A096162(n,k), read by rows.

1, 2, 2, 3, 6, 6, 4, 12, 12, 24, 24, 5, 20, 20, 60, 60, 120, 120, 6, 30, 30, 30, 120, 120, 120, 360, 360, 720, 720, 7, 42, 42, 42, 210, 210, 210, 210, 840, 840, 840, 2520, 2520, 5040, 5040, 8, 56, 56, 56, 56, 336, 336, 336, 336, 336, 1680, 1680, 1680, 1680, 1680, 6720, 6720, 6720, 20160, 20160

Offset: 1

Alford Arnold, Jun 21 2010

			A035206 begins 1 2 1 3 6 1 4 12 6 12 1 ...
A096162 begins 1 1 2 1 1 6 1 1 2 2 24 ...
therefore
A178888 begins 1 2 2 3 6 6 4 12 12 24 24 ...
with row sums 1,4,15,76,405,2616,18613,151432,1367649...A178887

Cf. A035206, A096162, A178887 (row sums).

A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 18, 12, 1, 5, 40, 60, 20, 1, 6, 75, 200, 150, 30, 1, 7, 126, 525, 700, 315, 42, 1, 8, 196, 1176, 2450, 1960, 588, 56, 1, 9, 288, 2352, 7056, 8820, 4704, 1008, 72, 1, 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1, 11, 550, 7425, 39600, 97020

Offset: 0

Paul Barry, Feb 03 2005

Columns include A000027, A002411, A004302, A108647, A134287. Row sums are C(2n+1,n+1) or A001700.
T(n-1,k-1) is the number of ways to put n identical objects into k of altogether n distinguishable boxes. See the partition array A035206 from which this triangle arises after summing over all entries related to partitions with fixed part number k.
T(n, k) is also the number of order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - Abdullahi Umar, Oct 02 2008
The o.g.f. of the (n+1)-th diagonal is given by G(n, x) = (n+1)*Sum_{k=1..n} A001263(n, k)*x^(k-1) / (1 - x)^(2*n+1), for n >= 1 and for n = 0 it is G(0, x) = 1/(1-x). - Wolfdieter Lang, Jul 31 2017

			The triangle T(n, k) begins:
n\k  0   1    2     3     4     5     6    7  8 9 ...
0:   1
1:   2   1
2:   3   6    1
3:   4  18   12     1
4:   5  40   60    20     1
5:   6  75  200   150    30     1
6:   7 126  525   700   315    42     1
7:   8 196 1176  2450  1960   588    56    1
8:   9 288 2352  7056  8820  4704  1008   72  1
9:  10 405 4320 17640 31752 26460 10080 1620 90 1
...  reformatted. - _Wolfdieter Lang_, Jul 31 2017
From _R. J. Mathar_, Mar 29 2013: (Start)
The matrix inverse starts
       1;
      -2,       1;
       9,      -6,      1;
     -76,      54,    -12,      1;
    1055,    -760,    180,    -20,   1;
  -21906,   15825,  -3800,    450, -30,   1;
  636447, -460026, 110775, -13300, 945, -42, 1; (End)
O.g.f. of 4th diagonal [4, 40,200, ...] is G(3, x) = 4*(1 + 3*x + x^2)/(1 - x)^7, from the n = 3 row [1, 3, 1] of A001263. See a comment above. - _Wolfdieter Lang_, Jul 31 2017

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 16.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv:1306.4628 [math.CO], 2013.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.

Cf. A008459, A122899, A194595, A197653.
Cf. A007318, A000894 (central terms), A132813 (mirrored).
Cf. A000027, A002411, A004302, A108647, A134287, A135278, A001263.

a103371 n k = a103371_tabl !! n !! k
a103371_row n = a103371_tabl !! n
a103371_tabl = map reverse a132813_tabl
-- Reinhard Zumkeller, Apr 04 2014

/* As triangle */ [[Binomial(n,n-k)*Binomial(n+1,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 01 2017

A103371 := (n,k) -> binomial(n,k)^2*(n+1)/(k+1);
seq(print(seq(A103371(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 19 2011

Flatten[Table[Binomial[n,n-k]Binomial[n+1,n-k],{n,0,10},{k,0,n}]] (* Harvey P. Dale, May 26 2014 *)
CoefficientList[Series[Series[E^(x(1+y))(BesselI[0,2*x*Sqrt[y]]+BesselI[1,2*x*Sqrt[y]]/Sqrt[y]),{x,0,8}],{y,0,8}],{x,y}]*Range[0,8]! (* Natalia L. Skirrow, Apr 14 2025 *)

create_list(binomial(n,k)*binomial(n+1,k+1),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n+1,k+1), ", "))) \\ G. C. Greubel, Nov 09 2018

Number triangle T(n, k) = C(n, n-k)*C(n+1, n-k) = C(n, k)*C(n+1, k+1); Column k of this triangle has g.f. Sum_{j=0..k} (C(k, j)*C(k+1, j) * x^(k+j))/(1-x)^(2*k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)*C(n+1, k).
T(n,k) = C(n, k)*Sum_{j=0..(n-k)} C(n-j, k). - Paul Barry, Jan 12 2006
T(n,k) = (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle (with offset [1,1]).
O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. of A001263, Narayana triangle). - Wolfdieter Lang, Nov 13 2007
From Peter Bala, Jan 24 2008: (Start)
Matrix product of A007318 and A122899.
O.g.f. for row n: (1-x)^n*P(n,1,0,(1+x)/(1-x)) = 1/(2*x)*(1-x)^(n+1)*( Legendre_P(n+1,(1+x)/(1-x)) - Legendre_P(n,(1+x)/(1-x)) ), where P(n,a,b,x) denotes the Jacobi polynomial.
O.g.f. for column k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. (End)
Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,1). (Cf. A194595, A197653, A197654). - Peter Luschny, Oct 20 2011
T(n,k) = A003056(n+1,k+1)*C(n,k)^2/(k+1). - Peter Luschny, Oct 29 2011
T(n,k) = A007318(n, k)*A135278(n, k), n >= k >= 0. - Wolfdieter Lang, Jul 31 2017
From Natalia L. Skirrow, Apr 14 2025: (Start)
T(n,k) = A008459(n,k) + n*N(n,k+1).
E.g.f.: e^(x*(1+y))*(I_0(2*x*sqrt(y)) + I_1(2*x*sqrt(y))/sqrt(y)), where I_n is the modified Bessel function of the first kind. (The I_0 contributes A008459(n,k), the I_1 contributes n*N(n,k+1))
O.g.f. for row n: (n+1)*2F1(-n,-n;2;y) = (n+1)*2F1(2+n,2+n;2;y)*(1-y)^(2*(n+1)) (by Euler's hypergeometric transformation); (n+1)*2F1(2+n,2+n;2;y) is the o.g.f. for row n of (k+n+1)!^2/(k!*(k+1)!*n!*(n+1)!), which is column n+1 of A132812.
O.g.f. for column k: 2F1(1+k,2+k;1;x)*x^k = 2F1(-k,-1-k;1;x)*x^k/(1-x)^(2+2*k). 2F1(-k,-1-k;1;x) is the kth row of A132813, the reflection of the kth row of this triangle.
O.g.f. for diagonal d (beginning at a(d,0)): (d+1)*x^d*2F1(d+1,d+2;2;x*y). 2F1(d+1,d+2;2;x) = 2F1(1-d,-d;2;x)/(1-x)^(2*d+1), numerator being the o.g.f. of row d of the Narayana triangle.
These respectively yield:
T(n,k) = Sum_{i=0..n+k} C(2*(n+1),i)*(-1)^i*A132812(n+1+k-i,n+1),
T(d+k,k) = Sum_{i=0..k} C(d-i+1+2*k,d-i)*T(k,k-i),
T(d+k,k) = Sum_{i=0..d} C(k-i + 2*d,k-i)*N(d,i+1)*(d+1).
E.g.f. for column k: 1F1(2+k;1;x)*x^k/k!.
E.g.f. for diagonal d: (d+1)*x^d*1F1(d+2;2;x*y)/d!. (End)

A049009 Number of functions from a set to itself such that the sizes of the preimages of the individual elements in the range form the n-th partition in Abramowitz and Stegun order.

Original entry on oeis.org

1, 1, 2, 2, 3, 18, 6, 4, 48, 36, 144, 24, 5, 100, 200, 600, 900, 1200, 120, 6, 180, 450, 300, 1800, 7200, 1800, 7200, 16200, 10800, 720, 7, 294, 882, 1470, 4410, 22050, 14700, 22050, 29400, 176400, 88200, 88200, 264600, 105840, 5040, 8, 448, 1568, 3136, 1960

Offset: 0

Alford Arnold

a(n,k) is a refinement of 1; 2,2; 3,18,6; 4,84,144,24; ... cf. A019575.
a(n,k)/A036040(n,k) and a(n,k)/A048996(n,k) are also integer sequences.
Apparently a(n,k)/A036040(n,k) = A178888(n,k). - R. J. Mathar, Apr 17 2011
Let f,g be functions from [n] into [n]. Let S_n be the symmetric group on n letters. Then f and g form the same partition iff S_nfS_n = S_ngS_n. - Geoffrey Critzer, Jan 13 2022

			Table begins:
  1;
  1;
  2,  2;
  3, 18,  6;
  4, 48, 36, 144, 24;
  ...
For n = 4, partition [3], we can map all three of {1,2,3} to any one of them, for 3 possible values. For n=5, partition [2,1], there are 3 choices for which element is alone in a preimage, 3 choices for which element to map that to and then 2 choices for which element to map the pair to, so a(5) = 3*3*2 = 18.

O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page38.

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A019575, A035206, A035796, A036038, A036040, A048996.

Row sizes A000041, sums A000312.

f[list_] := Multinomial @@ Join[{nn - Length[list]}, Table[Count[list, i], {i, 1, nn}]]*Multinomial @@ list; Table[nn = n; Map[f, IntegerPartitions[nn]], {n, 0, 10}] // Grid (* Geoffrey Critzer, Jan 13 2022 *)

C(sig)={my(S=Set(sig)); (binomial(vecsum(sig), #sig)) * (#sig)! * vecsum(sig)! / (prod(k=1, #S, (#select(t->t==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

a(n,k) = A036038(n,k) * A035206(n,k).

Better definition from Franklin T. Adams-Watters, May 30 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

A213941 Partition array a(n,k) with the total number of bracelets (D_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 12, 12, 24, 3, 5, 20, 40, 60, 120, 120, 12, 6, 30, 90, 45, 180, 720, 220, 600, 1440, 900, 60, 7, 42, 126, 168, 315, 1890, 1050, 1890, 2100, 12600, 6720, 6300, 18900, 7560, 360, 8, 56, 224, 280, 224, 672, 4032, 6384, 5544, 6384, 5880, 45360

Offset: 1

Wolfdieter Lang, Jul 20 2012

This array is obtained by multiplying the entry of the array A213939(n,k) (number of bracelets (dihedral D_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A213939(n,k)): a(n,k)=A213939(n,k)*A035206(n,k), k=1..p(n)=A000041(n), n>=1. The row sums then give the total number of bracelets with n beads from n colors, given by A081721(n).
See A212359 for references, the 'exponentiation', and a link. For multiset signatures and representative multisets defining color multinomials see also a link in A213938.
The corresponding triangle with the summed row entries related to partitions of n with fixed number of parts is A214306.

			n\k 1   2    3    4    5     6     7     8     9    10   11
1   1
2   2   1
3   3   6    1
4   4  12   12   24    3
5   5  20   40   60  120   120    12
6   6  30   90   45  180   720   220   600  1440   900   60
...
Row m=7 is: 7 42 126 168 315 1890 1050 1890 2100 12600 6720 6300 18900 7560 360.
For the rows n=1 to n=15 see the link.
a(3,1) = 3 because the 3 bracelets with 3 beads coming in 3 colors have the color multinomials (here monomials) c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is [3], the color representative is c[1]^3, and the equivalence class with color signature from the partition [3] has the three given members. There is no difference between necklace and bracelet numbers in this case.
a(3,2) = 6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A213939(3,2) = 1, the only 3-bracelet cyclic(112) (taking j for the color c[j]), and A035206(3,2) = 6 members of the whole color equivalence class: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332). There is no difference between necklaces and bracelets numbers in this case.
a(3,3) = 1, color signature 1^3 = 1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A213939(3,3)=1 from the bracelet cyclic(1,2,3). The necklace (1,3,2) becomes equivalent to this one under D_3 operation. There are no other members in this class (A035206(3,3)=1).
The sum of row No. 3 is 10 = A081721(3). The bracelets are 111, 222, 333, 112, 113, 221, 223, 331, 332 and 123, all taken cyclically.

Wolfdieter Lang, Rows n=1 to n=15.

Cf. A213939, A035206, A081721, A214306.

a(n,k) = A213939(n,k)*A035206(n,k), k=1, 2, ..., p(n) = A000041(n), n >= 1.

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 4, 1, 5, 10, 10, 10, 10, 5, 1, 6, 15, 15, 15, 20, 20, 20, 15, 15, 6, 1, 7, 21, 21, 21, 35, 35, 35, 35, 35, 35, 35, 21, 21, 7, 1, 8, 28, 28, 28, 28, 56, 56, 56, 56, 56, 70, 70, 70, 70, 70, 56, 56, 56, 28, 28, 8, 1, 9, 36, 36, 36, 36, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Alford Arnold, Sep 14 2004

A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).

Partitions are in Abramowitz and Stegun order. - Franklin T. Adams-Watters, Nov 20 2006

			A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
Table begins
.
1
2 1
3 3  1
4 6  6  4  1
5 10 10 10 10 5  1
6 15 15 20 15 20 15 20 15 6 1
.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.

Table[Sequence @@
  Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
  1, 10}] (* Olivier Gérard, May 07 2024 *)

a(n) = Combin( A036042(n), A036043(n) )

A212360 Partition array a(n,k) with the total number of necklaces (C_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

Original entry on oeis.org

1, 2, 1, 3, 6, 2, 4, 12, 12, 36, 6, 5, 20, 40, 120, 180, 240, 24, 6, 30, 90, 60, 300, 1200, 320, 1200, 2700, 1800, 120, 7, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 8, 56, 224, 392, 280, 1176, 7056, 11760, 9072, 11760, 11760, 88200, 58800, 176400, 22260, 58800, 470400, 352800, 141120, 529200, 141120, 5040

Offset: 1

Wolfdieter Lang, Jun 25 2012

This array is obtained by multiplying the entry of the array A212359(n,k) (number of necklaces (C_n symmetry) with n beads, each available in n colors, with color representative given by the n-multiset representative obtained from the k-th partition of n in A-St order after 'exponentiation') with the entry of the array A035206(n,k) (number of members in the equivalence class represented by the color multiset considered for A212359(n,k)):  a(n,k)=A212359(n,k)* A035206(n,k), k=1..p(n)= A000041(n), n>=1. The row sums then give the total number of necklaces with beads from n colors, given by A056665(n).
See A212359 for references, the 'exponentiation', and a link.
The corresponding triangle with the summed row entries which belong to partitions of n with fixed number of parts is A213935. [From Wolfdieter Lang, Jul 12 2012]

			n\k  1   2   3   4    5     6    7     8     9    10   11
1    1
2    2   1
3    3   6   2
4    4  12  12  36    6
5    5  20  40 120  180   240   24
6    6  30  90  60  300  1200  320  1200  2700  1800  120
...
See the link for the rows n=1..15.
a(3,1)=3 because the 3 necklaces with 3 beads coming in 3 colors have the color multinomials (here monomials)  c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is 3, the color representative is c[1]^3, and the equivalence class with color signature from the partition 3 has the three given members.
a(3,2)=6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A212359(3,2)=1, the only 3-necklace cyclic(112) (taking j for the color  c[j]), and  A035206(3,2)=6 members of the whole color equivalence class: cyclic(112), cyclic(113),  cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3)=2, color signature 1^3=1,1,1 with representative multinomial  c[1]*c[2]*c[3] with coefficient A212359(3,3)=2 from the two necklaces cyclic(1,2,3) and cyclic (1,3,2). There are no other members in this class (A035206(3,3)=1).
The sum of row nr. 3 is 11=A056665(3). See the example given there with c[1]=R, c[2]=G and c[3]=B.

Wolfdieter Lang, Rows n=1..15.

Cf. A212359, A035206, A056665, A213935.

a(n,k) = A212359(n,k)*A035206(n,k), k=1,2,...,p(n)= A000041(n), n>=1.

A214312 a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.

Original entry on oeis.org

3, 120, 2040, 21420, 183330, 1320480, 8691480, 52727400, 303958710, 1674472800, 8928735816, 46280581620, 234611247780, 1166708558400, 5710351190400, 27565250985360, 131495088522060, 620771489730000, 2903870526350640, 13473567673441260, 62061657617625204, 283995655732351200

Offset: 4

Wolfdieter Lang, Jul 31 2012

nonn

This is the fourth column (m=4) of triangle A214306.
Each 4 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say  [p[1], p[2], p[3], p[4]], with p[1] >= p[2]  >= p[3] >= p[4] >= 1, there are A213941(n,k)= A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,4)= A008284(n,4)  partitions of n with 4 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this with A032275 where also bracelets with less than four colors are included, and the color repertoire is only [c[1], c[2], c[3], c[4]] for all n.

			a(5) = A213941(5,6) = 120 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11234, 11243, 11324, 12134, 13124 and 14123, all six taken cyclically, each representing a class of order A035206(5,6) = 20 (if all 5 colors are used). For example, cyclic(11342) becomes equivalent to cyclic(11243) by turning over or reflection. The multiplicity 20 depends only on the color signature.

Andrew Howroyd, Table of n, a(n) for n = 4..100

Cf. A213941, A214306, A214309 (m=4, representative bracelets), A214313 (m=5).

t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
a56344[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := Binomial[n, 4]*a56344[n, 4];
Table[a[n], {n, 4, 25}] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

a(n) = A214306(n,4), n >= 4.
a(n) = sum(A213941(n,k),k = A214314(n,4) .. (A214314(n,4) - 1 + A008284(n,4))), n >= 4.
a(n) = binomial(n,4) * A056344(n). - Andrew Howroyd, Mar 25 2017

A098545 Row sums of A098546.

Original entry on oeis.org

1, 3, 7, 21, 51, 148, 365, 983, 2461, 6360, 15687, 39757, 97033, 240425, 582622, 1421273, 3409861, 8222920, 19565707, 46680362, 110309476, 260876036, 612293443, 1437616751, 3354111156, 7823501148, 18157700800, 42112132458

Offset: 1

Alford Arnold, Sep 14 2004

By using multisets (cf. A001700) and multinomials (cf. A005651); A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end (compare A090657).

			A098546 begins
1
1 2
1 3 3
1 4 6 6 4
so sequence begins 1 3 7 21 ...

Cf. A000041, A000312.

a(n) = Sum_{k=1..n} binomial(n, k)*A008284(n, k). - Vladeta Jovovic, Jul 24 2005

More terms from Vladeta Jovovic, Jul 24 2005

A214310 a(n) is the number of all three-color bracelets (necklaces with turning over allowed) with n beads and the three colors are from a repertoire of n distinct colors, for n >= 3.

Original entry on oeis.org

1, 24, 180, 1120, 5145, 23016, 91056, 357480, 1327095, 4893680, 17525508, 62254920, 217457695, 753332160, 2581110000, 8779264032, 29624681763, 99350001360, 331159123260, 1098168382080, 3624003213369, 11908069219816, 38972450763000, 127087400895000

Offset: 3

Wolfdieter Lang, Jul 31 2012

nonn

This is the third column (m=3) of triangle A214306.
Each 3 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], p[3]], with p[1] >= p[2] >= p[3] >= 1, there are A213941(n,k)= A035206(n,k)* A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,3)= A008284(n,3) partitions of n with 3 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
Compare this with A027671 where also single color bracelets are included, and the color repertoire is only [c[1], c[2], c[3]] for all n.

			a(5) = A213941(5,4) + A213941(5,5) = 60 + 120 = 180 from the bracelet (with colors j for c[j], j=1, 2, ..., 5) 11123 and 11213, both taken cyclically, each representing a class of order A035206(5,4)= 30 (if all 5 colors are used), and 11223, 11232, 12123 and 12213, all taken cyclically, each representing a class of order A035206(5,5)= 30. For example, cyclic(11322) becomes equivalent to cyclic(11223) by turning over or reflection. The multiplicity A035206 depends only on the color signature.

Andrew Howroyd, Table of n, a(n) for n = 3..100

Cf. A213941, A214306, A214307 (m=3, representative bracelets), A214312 (m=4).

a(n) = A214306(n,3), n >= 3.
a(n) = sum(A213941(n,k), k = A214314(n,3).. (A214314(n,3) - 1 + A008284(n,3))), n >= 3.
a(n) = binomial(n,3) * A056343(n). - Andrew Howroyd, Mar 25 2017

a(26) from Andrew Howroyd, Mar 25 2017

A214313 a(n) is the number of all five-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 5.

Original entry on oeis.org

12, 900, 25200, 442680, 5846400, 64420272, 622175400, 5466166200, 44611306740, 343916472900, 2531921456064, 17956666859040, 123458676825120, 827056125453600, 5419508203393200, 34847210197637424, 220424306985639540, 1374479672119161300, 8463477229726134000, 51536194734146965920, 310706598354410079360

Offset: 5

Wolfdieter Lang, Aug 08 2012

nonn

This is the fifth column (m=5) of triangle A214306.
Each 5 part partition of n, with the parts written in nonincreasing order, defines a color signature. For a given color signature, say [p[1], p[2], ..., p[5]], with p[1] >= p[2] >= .. >=  p[5] >= 1, there are A213941(n,k) = A035206(n,k)*A213939(n,k) bracelets if this signature corresponds (with the order of the parts reversed) to the k-th partition of n in Abramowitz-Stegun (A-St) order. See A213941 for more details. Here all p(n,5)= A008284(n,5) partitions of n with 5 parts are considered. The color repertoire for a bracelet with n beads is [c[1], ..., c[n]].
It appears that this sequence is divisible by 12, producing  1, 75, 2100, 36890, 487200, 5368356, 51847950, 455513850, ...
Compare this with A056345 where only 5 colors are used for all n >= 5.

			a(6) = A213941(6,10) = 900 from the bracelet with color signature [2,1,1,1,1] and color repertoire [c[j], j=1, 2, ..., 6]. There are A213939(6,10) = 30 bracelets with representative color multinomials c[1]^2 c[2] c[3] c[4] c[5]. If the colors c[j] are taken as j, e.g., 112345, 112354, 112435, 112453, 112534, 112543, 113245, 113254, 113425, (113452 is equivalent to 112543 by turning over), 113524, (113542 ==112453), 114235, ..., 121345, ... (all taken cyclically). Each of these 30 bracelets represents a class of A035206(6,10) = 30 bracelets when all six colors are used. Thus a(6) = 30*30 = 900 = 12*75.

Andrew Howroyd, Table of n, a(n) for n = 5..100

Cf. A213941, A214306, A214311 (m=5, representative bracelets), A214312 (m=4).

a(n) = A214306(n,5), n >= 5.
a(n) = sum(A213941(n,k),k = A214314(n,5) .. (A214314(n,5) - 1 + A008284(n,5))), n >= 5.
a(n) = binomial(n,5) * A056345(n). - Andrew Howroyd, Mar 25 2017

cp's OEIS Frontend

A178888 Irregular table A035206(n,k)*A096162(n,k), read by rows.

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A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).

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A049009 Number of functions from a set to itself such that the sizes of the preimages of the individual elements in the range form the n-th partition in Abramowitz and Stegun order.

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A213941 Partition array a(n,k) with the total number of bracelets (D_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

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A212360 Partition array a(n,k) with the total number of necklaces (C_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.

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A214312 a(n) is the number of all four-color bracelets (necklaces with turning over allowed) with n beads and the four colors are from a repertoire of n distinct colors, for n >= 4.

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A098545 Row sums of A098546.

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