A036074 -id:A036074 - cp's OEIS frontend

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

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

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

A036078 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.

Original entry on oeis.org

1, 2, 13, 127, 1508, 20859, 332557, 6019108, 121462267, 2692076295, 64846340130, 1684713690917, 46916754353013, 1393010598959594, 43889040801834505, 1461369418905803027, 51243270154712083052

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.
Index entries for sequences related to sorting

Cf. A001861, A002872-A002875, A036074, ...

mx = 16; p = 8; 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] * 8^k * BellB[k, 1/8] * 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=8. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (8*n/LambertW(8*n))^n * exp(n/LambertW(8*n) + (8*n/LambertW(8*n))^(1/8) - n - 9/8) / sqrt(1 + LambertW(8*n)). - Vaclav Kotesovec, Jul 10 2022

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

A036079 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.

Original entry on oeis.org

1, 2, 14, 150, 1942, 29174, 505318, 9957798, 219177942, 5303780758, 139554619206, 3962202725254, 120644298135478, 3918518255860342, 135117086088186662, 4925731652244913766, 189170325211554345366, 7629758975467859662678, 322296334808561664346886

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.
Index entries for sequences related to sorting

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

mx = 16; p = 9; 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] * 9^k * BellB[k, 1/9] * 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=9. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (9*n/LambertW(9*n))^n * exp(n/LambertW(9*n) + (9*n/LambertW(9*n))^(1/9) - n - 10/9) / sqrt(1 + LambertW(9*n)). - Vaclav Kotesovec, Jul 10 2022

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

A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.

Original entry on oeis.org

1, 2, 15, 175, 2452, 39703, 741177, 15771270, 375485507, 9837064575, 280338965720, 8623355105347, 284589703065137, 10022926411599482, 374900187362983015, 14830483377507515247, 618219446355189917804, 27071966121397255354079, 1241912851303663452150377

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.
Index entries for sequences related to sorting

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

mx = 16; p = 10; 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] * 10^k * BellB[k, 1/10] * 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=10. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (10*n/LambertW(10*n))^n * exp(n/LambertW(10*n) + (10*n/LambertW(10*n))^(1/10) - n - 11/10) / sqrt(1 + LambertW(10*n)). - Vaclav Kotesovec, Jul 10 2022

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

A036082 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=12.

Original entry on oeis.org

1, 2, 17, 231, 3724, 68819, 1464781, 35645040, 973624491, 29313919207, 960689482494, 33997330377817, 1291521482389621, 52395164853506674, 2259005857941805253, 103064324686839195035, 4957382457319437575820, 250592665906288206715951, 13275467282249493427541201

Offset: 0

N. J. A. Sloane

nonn

In general, for p>=2, a(n) ~ c * (p*n/LambertW(p*n))^n * exp(n/LambertW(p*n) + (p*n/LambertW(p*n))^(1/p) - n - 1 - 1/p) / sqrt(1 + LambertW(p*n)), where c = 1 for p>=3 and c = exp(-1/4) for p=2. - Vaclav Kotesovec, Jul 10 2022

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.
Index entries for sequences related to sorting

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

mx = 16; p = 12; 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] * 12^k * BellB[k, 1/12] * 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=12. - Vaclav Kotesovec, Jul 03 2022

a(n) ~ (12*n/LambertW(12*n))^n * exp(n/LambertW(12*n) + (12*n/LambertW(12*n))^(1/12) - n - 13/12) / sqrt(1 + LambertW(12*n)). - Vaclav Kotesovec, Jul 10 2022

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

cp's OEIS Frontend

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

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

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

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

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A036078 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=8.

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A036079 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=9.

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A036080 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=10.

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A036082 E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=12.