A002872 -id:A002872 - cp's OEIS frontend

A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

1, 2, 9, 55, 412, 3619, 36333, 408888, 5080907, 68914023, 1011165446, 15935379409, 268125052373, 4792458452162, 90605469012877, 1805135197261131, 37775862401203916, 827992670793489263

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036076, ...

mx = 16; p = 4; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 4^k * BellB[k, 1/4] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n):=sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n); /* Vladimir Kruchinin, Sep 14 2010 */

a(n) = sum(sum(binomial(m,i)*sum(binomial(i,j)*(1/4)^j*(3*j+i)^n,j,0,i)*(-5/4)^(m-i),i,0,m)/m!,m,1,n), n > 0. - Vladimir Kruchinin, Sep 14 2010
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=4. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (4*n/LambertW(4*n))^n * exp(n/LambertW(4*n) + (4*n/LambertW(4*n))^(1/4) - n - 5/4) / sqrt(1 + LambertW(4*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters

A080337 Bisection of A080107.

Original entry on oeis.org

1, 3, 12, 59, 339, 2210, 16033, 127643, 1103372, 10269643, 102225363, 1082190554, 12126858113, 143268057587, 1778283994284, 23120054355195, 314017850216371, 4444972514600178, 65435496909148513, 999907522895563403, 15832873029742458796, 259377550023571768075

Offset: 1

Wouter Meeussen, Mar 18 2003

nonn

Number of symmetric positions of non-attacking rooks on upper-diagonal part of 2n X 2n chessboard.
Number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=2+max(prefix) for k>=1, see example. - Joerg Arndt, Apr 25 2010
Number of achiral color patterns in a row or loop of length 2n-1. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 24 2018
Stirling transform of A005425(n-1) per Knuth reference. - Robert A. Russell, Apr 28 2018

			From _Joerg Arndt_, Apr 25 2010: (Start)
For n=0 there is one empty string (term a(0)=0 not included here); for n=1 there is one string [0]; for n=2 there are 3 strings [00], [01], and [02];
for n=3 there are a(3)=12 strings (in lexicographic order):
01: [000],
02: [001],
03: [002],
04: [010],
05: [011],
06: [012],
07: [013],
08: [020],
09: [021],
10: [022],
11: [023],
12: [024].
(End)
For a(3) = 12, both the row and loop patterns are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE. - _Robert A. Russell_, Apr 24 2018

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765). - Robert A. Russell, Apr 28 2018

Alois P. Heinz, Table of n, a(n) for n = 1..514
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.4, pp. 364-366.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 18, 29.
J. Quaintance, Letter representations of rectangular m x n x p proper arrays, arXiv:math/0412244 [math.CO], 2004-2006.

Row sums of A140735.
Cf. A002872, A080107.
Column k=2 of A305962.

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+2))
    end:
a:= n-> b(n, -1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 15 2018

Table[Sum[ Binomial[n, k] A002872[[k + 1]], {k, 0, n}], {n, 0, 24}]
Aodd[m_, k_] := Aodd[m, k] = If[m > 1, k Aodd[m-1, k] + Aodd[m-1, k-1]
  + Aodd[m-1, k-2], Boole[m==1 && k==1]]
Table[Sum[Aodd[m, k], {k, 1, 2m-1}], {m, 1, 30}] (* Robert A. Russell, Apr 24 2018 *)
x[n_] := x[n] = If[n<2, n+1, 2x[n-1] + (n-1) x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[n, k] x[k-1], {k, 0, n}], {n, 30}] (* Robert A. Russell, Apr 28 2018, after Knuth reference *)

x='x+O('x^66);
egf=exp(x+exp(x)+exp(2*x)/2-3/2); /* = 1 +3*x +6*x^2 +59/6*x^3 +113/8*x^4 +... */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 29 2011 */

Binomial transform of A002872 (sorting numbers).
E.g.f.: exp(x+exp(x)+exp(2*x)/2-3/2) = exp(x+sum(j=1,2, (exp(j*x)-1)/j ) ). - Joerg Arndt, Apr 29 2011
From Robert A. Russell, Apr 24 2018: (Start)
Aodd[n,k] = [n>1]*(k*Aodd[n-1,k]+Aodd[n-1,k-1]+Aodd[n-1,k-2])+[n==1]*[k==1]
a(n) = Sum_{k=1..2n-1} Aodd[n,k]. (End)
a(n) = Sum_{k=0..n} Stirling2(n, k)*A005425(k-1). (from Knuth reference) - Robert A. Russell, Apr 28 2018

Comment corrected by Wouter Meeussen, Aug 14 2009

A084708 Number of set partitions up to rotations and reflections.

Original entry on oeis.org

1, 2, 3, 7, 12, 37, 93, 354, 1350, 6351, 31950, 179307, 1071265, 6845581, 46162583, 327731950, 2437753740, 18948599220, 153498350745, 1293123243928, 11306475314467, 102425554299516, 959826755336242, 9290811905391501

Offset: 1

Wouter Meeussen, Jul 02 2003

nonn

Combines the symmetry operations of A080107 and A084423.

Equivalently, number of n-bead bracelets using any number of unlabeled (interchangable) colors. - Andrew Howroyd, Sep 25 2017

			SetPartitions[6] is the first to decompose differently from A084423: 4 cycles of length 1, 2 of 2, 9 of 3, 16 of 6, 6 of 12.
a(7) = 1 + A056357(7) + A056358(7) + A056359(7) + A056360(7) + A056361(7) + 1 = 1 + 8 + 31 + 33 + 16 + 3 + 1 = 93.

Michael De Vlieger, Table of n, a(n) for n = 1..577
Colin Adams, Chaim Even-Zohar, Jonah Greenberg, Reuben Kaufman, David Lee, Darin Li, Dustin Ping, Theodore Sandstrom, and Xiwen Wang, Virtual Multicrossings and Petal Diagrams for Virtual Knots and Links, arXiv:2103.08314 [math.GT], 2021.
Tilman Piesk, Partition related number triangles
N. J. A. Sloane, Generating functions [From _Wouter Meeussen_, Dec 06 2008]

Cf. A080107, A084423, A080510, A002872, A002874, A141003, A036075, A141004, A036077, A152176.

<A080107 *); Table[{Length[ # ], First[ # ]}&/@ Split[Sort[Length/@Split[Sort[First[Sort[Flatten[ {#, Map[Sort, (#/. i_Integer:>w+1-i), 2]}& @(NestList[Sort[Sort/@(#/. i_Integer :> Mod[i+1, w, 1])]&, #, w]), 1]]]&/@SetPartitions[w]]]]], {w, 1, 10}]
u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; a[n_]:=1/n*Plus@@(EulerPhi[ # ]u[Quotient[n,# ],# ]&/@Divisors[n]); Table[a[n]/2+If[EvenQ[n],u[n/2,2],Sum[Binomial[n/2-1/2,k] u[k,2], {k,0,n/2-1/2}]]/2,{n,40}] (* Wouter Meeussen, Dec 06 2008 *)

a(n) = (A080107(n)+A084423(n))/2. - Wouter Meeussen and Vladeta Jovovic, Nov 28 2008

a(12) from Vladeta Jovovic, Jul 15 2007

More terms from Vladeta Jovovic's formula given in Mathematica line. - Wouter Meeussen, Dec 06 2008

A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.

Original entry on oeis.org

1, 2, 10, 70, 602, 6078, 70402, 917830, 13253002, 209350350, 3584098770, 66012131222, 1300004931162, 27232369503902, 604103160535330, 14136908333006822, 347827448896896554, 8971450949011952494

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872-A002875, A036074.
u[n,j] generates for j=1, A000110 Bell numbers; j=2, A002872; j=3, A002874; j=4, A141003 (Mathar); j=5, this sequence; j=6, A141004 (Mathar); j=7, A036077. - Wouter Meeussen, Dec 06 2008
Column 5 of A162663.

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,5],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 5; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 5^k * BellB[k, 1/5] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=5.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=5. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (5*n/LambertW(5*n))^n * exp(n/LambertW(5*n) + (5*n/LambertW(5*n))^(1/5) - n - 6/5) / sqrt(1 + LambertW(5*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A036077 The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.

Original entry on oeis.org

1, 2, 12, 106, 1144, 14434, 209736, 3451290, 63194936, 1269555762, 27700698344, 651497885482, 16414347638936, 440651469115394, 12546081858835528, 377328994871025210, 11946046637611280120

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872-A002875, A036074.
u[n,j] generates for j=1, A000110; j=2, A002872; j=3, A002874; j=4, A141003; j=5, A036075; j=6, A141004; j=7, this sequence. - Wouter Meeussen, Dec 06 2008
Column 7 of A162663.

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,7],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 7; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 7^k * BellB[k, 1/7] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=7.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=7. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (7*n/LambertW(7*n))^n * exp(n/LambertW(7*n) + (7*n/LambertW(7*n))^(1/7) - n - 8/7) / sqrt(1 + LambertW(7*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A141004 Expansion of e.g.f. exp(Sum_{d|6} (exp(d*x)-1)/d).

Original entry on oeis.org

1, 4, 28, 258, 2892, 37778, 560124, 9256010, 168182044, 3325057826, 70934634236, 1621828212826, 39517131361884, 1021237022557682, 27877344103738940, 800976143703407210, 24148078430008534428, 761815206361252780098, 25087729474993723079548

Offset: 0

R. J. Mathar, Jul 11 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

u[n,j] generates for j=1, A000110 Bell numbers; j=2, A002872 "Sorting numbers"; j=3, A002874 "Sorting numbers"; j=4, A141003 (Mathar); j=5, A036075 "Sorting numbers"; j=6, A141004 (Mathar); j=7, A036077 "Sorting numbers". - Wouter Meeussen, Dec 06 2008

Column k=6 of A162663.

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,6],{n,0,18}] (* Wouter Meeussen, Dec 06 2008 *)

A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 0, 1, 1, 3, 10, 12, 3, 9, 3, 0, 1, 1, 7, 33, 59, 30, 67, 42, 6, 18, 4, 0, 1, 1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1, 1, 31, 333, 1187, 1365, 3112, 3675, 1596, 2700, 1655, 330, 605, 195, 15, 45, 6, 0, 1

Offset: 1

Andrew Howroyd, Oct 24 2017

T(n,k) = coefficient of x^k for A(3,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 13 2018

			Triangle begins:
  1,  0,   1;
  1,  1,   3,   2,   0,   1;
  1,  3,  10,  12,   3,   9,   3,   0,   1;
  1,  7,  33,  59,  30,  67,  42,   6,  18,   4,  0,  1;
  1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1;
  ...
Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The second row is [1, 1, 3, 2, 0, 1] because the set partitions that are invariant under this permutation in increasing order of number of parts are {{1, 2, 3, 4, 5, 6}}; {{1, 2, 3}, {4, 5, 6}}; {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}; {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1395
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums are A002874.
Column k=3 gives A053156.
Maximum row values are A294202.
Unrelated to A002875.
Cf. A002872, A002873, A293181.

T:= proc(n, k) option remember; `if`([n, k]=[0, 0], 1, 0)+
     `if`(n>0 and k>0, k*T(n-1, k)+T(n-1, k-1)+T(n-1, k-3), 0)
    end:
seq(seq(T(n, k), k=1..3*n), n=1..8);  # Alois P. Heinz, Sep 20 2019

T[n_, k_] := T[n,k] = If[n>0 && k>0, k T[n-1,k] + T[n-1,k-1] + T[n-1,k-3], Boole[n==0 && k==0]] (* modification of Gilbert & Riordan recursion *)
Table[T[n, k], {n,1,10}, {k,1,3n}] // Flatten (* Robert A. Russell, Jun 13 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k)={my(ci=PermCycleIndex(CylinderPerms(3,n)[2])); StructsByCycleIndex(ci,k) - if(k>1,StructsByCycleIndex(ci,k-1))}
for (n=1, 6, for(k=1, 3*n, print1(T(n,k), ", ")); print);

G(n)={Vec(-1+serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
{ my(A=G(6)); for(n=1, #A, print(Vecrev(A[n]/y))) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = [n==0 & k==0] + [n>0 & k>0] * (k*T(n-1,k) + T(n-1,k-1) + T(n-1,k-3)). - Robert A. Russell, Jun 13 2018

T(n,k) = n!*[x^n*y^k] exp(Sum_{d|3} y^d*(exp(d*x) - 1)/d). - Andrew Howroyd, Sep 20 2019

A085483 Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.

Original entry on oeis.org

2, 2, 5, 2, 15, 14, 2, 35, 84, 43, 2, 75, 350, 430, 142, 2, 155, 1260, 2795, 2130, 499, 2, 315, 4214, 15050, 19880, 10479, 1850, 2, 635, 13524, 73143, 149100, 132734, 51800, 7193, 2, 1275, 42350, 334110, 987042, 1320354, 854700, 258948, 29186, 2, 2555, 130620, 1466515, 6038550, 11390673, 10878000, 5394750, 1313370, 123109

Offset: 1

James East, Aug 15 2003

			S_B(2,2)=5 because the relevant partitions of {-2,-1,1,2} are: {-2|-1|1|2}, {-1,1|-2|2}, {-1|1|-2,2}, {-1,1|-2,2}, {1,-2|-1,2}.
Triangle begins:
  2;
  2,   5;
  2,  15,   14;
  2,  35,   84,   43;
  2,  75,  350,  430,  142;
  2, 155, 1260, 2795, 2130, 499;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 4.
Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.

Cf. A008277, A005425.

S_B(n, 1) + ... + S_B(n, n) = A002872(n).

nn = 10; f[n_] := Sum[2^(n - 3 k) n!/((n - 2 k)! k!), {k, 0, n}]; Do[f[n], {n, 0, nn}]; Table[f[k]*StirlingS2[n, k], {n, nn}, {k, n}] (* Michael De Vlieger, Sep 21 2022, after Robert G. Wilson v at A005425 *)

A partition of {-n, ..., -1, 1, ..., n} into nonempty subsets X_1, ..., X_r is called "symmetric" if for each i -X_i = X_j for some j. S_B(n, k) is the number of such symmetric partitions whose induced partition on {1, ..., n} involves k nonempty subsets. S_B(n, k) = S(n, k) * a(k), where S(n, k) is A008277 and a(k) is A005425.

A036076 Expansion of e.g.f. exp((exp(p*x)-p-1)/p+exp(x)) for p=6.

Original entry on oeis.org

1, 2, 11, 87, 844, 9599, 125545, 1854234, 30407763, 546409567, 10654642428, 223763443039, 5030118977041, 120393730088818, 3054106291046267, 81792080931311015, 2304639285452820684, 68117438479292896255

Offset: 0

N. J. A. Sloane

nonn

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.
T. S. Motzkin, Sorting numbers ...: for a link to an annotated scanned version of this paper see A000262.

Robert Israel, Table of n, a(n) for n = 0..437
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074, ...

egf:=  exp((exp(6*x)-6-1)/6+exp(x)):
S:= series(egf,x,501):
seq(coeff(S,x,i)*i!, i=0..20); # Robert Israel, Nov 27 2022

mx = 16; p = 6; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 6^k * BellB[k, 1/6] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=6. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (6*n/LambertW(6*n))^n * exp(n/LambertW(6*n) + (6*n/LambertW(6*n))^(1/6) - n - 7/6) / sqrt(1 + LambertW(6*n)). - Vaclav Kotesovec, Jul 10 2022

Edited by N. J. A. Sloane, Jul 11 2008 at the suggestion of Franklin T. Adams-Watters.

A036081 The number of partitions of {1..(11n)} that are invariant under a permutation consisting of n 11-cycles.

Original entry on oeis.org

1, 2, 16, 202, 3044, 52794, 1055260, 24081754, 615896308, 17347970202, 531721375308, 17595339114554, 624882463734756, 23691503493287738, 954301756159098172, 40665568780962213530, 1826521141853468785364

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872, A002873, A002874, A002875, A036074.

Column 11 of A162663.

u[0, j_] := 1; u[k_, j_] := u[k, j] = Sum[Binomial[k-1, i-1]Plus@@(u[k-i, j]#^(i-1)&/@Divisors[j]), {i, k}]; Table[u[n, 11], {n, 0, 30}] (* Vincenzo Librandi, Dec 12 2012 - after Wouter Meeussen in similar sequences *)
mx = 16; p = 11; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 11^k * BellB[k, 1/11] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=11.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=11. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (11*n/LambertW(11*n))^n * exp(n/LambertW(11*n) + (11*n/LambertW(11*n))^(1/11) - n - 12/11) / sqrt(1 + LambertW(11*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

cp's OEIS Frontend

A036074 Expansion of e.g.f. exp((exp(p*x) - p - 1)/p + exp(x)) for p=4.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A080337 Bisection of A080107.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A084708 Number of set partitions up to rotations and reflections.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A036077 The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A141004 Expansion of e.g.f. exp(Sum_{d|6} (exp(d*x)-1)/d).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple