A002872 -id:A002872 - cp's OEIS frontend

A086365 n-th Bell number of type D: Number of symmetric partitions of {-n,...,n}\{0} such that none of the subsets is of the form {j,-j}.

1, 4, 15, 75, 428, 2781, 20093, 159340, 1372163, 12725447, 126238060, 1332071241, 14881206473, 175297058228, 2169832010759, 28136696433171, 381199970284620, 5383103100853189, 79065882217154085, 1205566492711167004, 19049651311462785947

Offset: 0

James East, Sep 04 2003

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. a(n) is the number of such symmetric partitions such that none of the X_i are of the form {j,-j}.

			a(2)=4 because the relevant partitions of {-2,-1,1,2} are {-2|-1|1|2}, {-2,-1|1,2}, {-2,1|-1,2} and {-2,-1,1,2}.

Cf. A002872, A086364.

x = 'x + O('x^16);
egf = -1 + exp(-x+sum(j=1,2,(exp(j*x)-1)/j))
/* egf == +x +2*x^2 +5/2*x^3 +25/8*x^4 +... (i.e., for offset 1) */
Vec( serlaplace(egf) )
/* Joerg Arndt, Apr 29 2011 */

E.g.f. (for offset 1): -1 + exp(-x + Sum_{j=1..2} (exp(j*x)-1)/j). - Joerg Arndt, Apr 29 2011

More terms from Joerg Arndt, Apr 29 2011

Definition shortened by M. F. Hasler, Oct 21 2012

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

A086365 n-th Bell number of type D: Number of symmetric partitions of {-n,...,n}\{0} such that none of the subsets is of the form {j,-j}.

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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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Mathematica

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

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Mathematica

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Extensions