A086364 -id:A086364 - 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

A135593 Number of n X n symmetric (0,1)-matrices with exactly n+1 entries equal to 1 and no zero rows or columns.

Original entry on oeis.org

2, 9, 36, 140, 540, 2142, 8624, 35856, 152280, 666380, 2982672, 13716144, 64487696, 310693320, 1528801920, 7691652992, 39474925344, 206758346256, 1103332900160, 5999356762560, 33197323465152, 186925844947424, 1069977071943936

Offset: 2

Vladeta Jovovic, Feb 25 2008

Vincenzo Librandi, Table of n, a(n) for n = 2..200

Cf. A000085, A055602, A086364.

A135593 := proc(n) n!*coeftayl( x^2*(x+2)/2*exp(x*(x+2)/2),x=0,n) ; end: seq(A135593(n),n=2..40) ; # R. J. Mathar, Mar 31 2008

Rest[Rest[CoefficientList[Series[x^2*(x+2)/2*E^(x*(x+2)/2), {x, 0, 20}], x]* Range[0, 20]!]] (* Vaclav Kotesovec, Oct 20 2012 *)
Flatten[{2,9,RecurrenceTable[{(n-5)*(n-2)*a[n]==(n-6)*n*a[n-1]+(n-4)*(n-1)*n*a[n-2],a[4]==36,a[5]==140},a,{n,4,20}]}] (* Vaclav Kotesovec, Oct 20 2012 *)

E.g.f.: x^2*(x+2)/2*exp(x*(x+2)/2).
Recurrence (for n>5): (n-5)*(n-2)*a(n) = (n-6)*n*a(n-1) + (n-4)*(n-1)*n*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 1/4*sqrt(2)*exp(sqrt(n)-n/2-1/4)*n^(n/2+3/2). - Vaclav Kotesovec, Oct 20 2012

More terms from R. J. Mathar, Mar 31 2008

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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